extreme f(x,y)=-e^{x-y^2+xy}

{ "query": { "display": "extreme points $$f\\left(x,\\:y\\right)=-e^{x-y^{2}+xy}$$", "symbolab_question": "FUNCTION#extreme f(x,y)=-e^{x-y^{2}+xy}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "extreme", "default": "\\mathrm{Saddle}(-2,-1)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Extreme Points of $$-e^{x-y^{2}+xy}:{\\quad}$$Saddle$$\\left(-2,\\:-1\\right)$$", "steps": [ { "type": "definition", "title": "Second Partial Derivative Test definition", "text": "Suppose that $$\\left(x,\\:y\\right)=\\left(a,\\:b\\right)\\:$$is a critical point of $$f\\left(x,\\:y\\right)$$<br/>and $$D\\left(x,\\:y\\right)=\\frac{\\partial^{2}f}{\\partial\\:x^{2}}\\frac{\\partial^{2}f}{\\partial\\:y^{2}}-\\left(\\frac{\\partial^{2}f}{\\partial\\:x\\partial\\:y}\\right)^{2}.\\:$$Then, <br/>If $$D\\left(a,\\:b\\right)\\:>\\:0\\:$$and $$\\frac{\\partial^{2}f}{\\partial\\:x^{2}}<0\\:$$then the point $$\\left(a,\\:b\\right)\\:$$is a local maximum.<br/>If $$D\\left(a,\\:b\\right)\\:>\\:0\\:$$and $$\\frac{\\partial^{2}f}{\\partial\\:x^{2}}>0\\:$$then the point $$\\left(a,\\:b\\right)\\:$$ is a local minimum.<br/>If $$D\\left(a,\\:b\\right)<0\\:$$then the point $$\\left(a,\\:b\\right)\\:$$ is a saddle point.<br/>If $$D\\left(a,\\:b\\right)=0\\:$$then test failed and the point $$\\left(a,\\:b\\right)\\:$$can be a local maximum, local minimum, saddle or neither." }, { "type": "interim", "title": "Find the critical points:$${\\quad}\\left(-2,\\:-1\\right)$$", "input": "-e^{x-y^{2}+xy}", "steps": [ { "type": "definition", "title": "Critical point definition", "text": "Critical points are points where the function is defined and its gradient is zero vector or undefined" }, { "type": "interim", "title": "Find where $$∇f\\left(x,\\:y\\right)$$ is equal to zero or undefined", "input": "-e^{x-y^{2}+xy}", "result": "\\left(-2,\\:-1\\right)", "steps": [ { "type": "interim", "title": "$$∇f\\left(x,\\:y\\right)=\\left(-e^{x-y^{2}+xy}\\left(y+1\\right),\\:-e^{x-y^{2}+xy}\\left(-2y+x\\right)\\right)$$", "input": "f=-e^{x-y^{2}+xy}", "steps": [ { "type": "step", "primary": "$$∇f\\left(x,\\:y\\right)=\\left(\\frac{\\partial\\:f}{\\partial\\:x},\\:\\frac{\\partial\\:f}{\\partial\\:y}\\right)$$" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(-e^{x-y^{2}+xy}\\right)=-e^{x-y^{2}+xy}\\left(y+1\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(-e^{x-y^{2}+xy}\\right)", "steps": [ { "type": "step", "primary": "Treat $$y\\:$$as a constant" }, { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-\\frac{\\partial\\:}{\\partial\\:x}\\left(e^{x-y^{2}+xy}\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(e^{x-y^{2}+xy}\\right)", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=x-y^{2}+xy$$" ], "result": "=\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "input": "\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "result": "=e^{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnor/Dc3XNYBnT8IDMcV0N5HI5S0StY1FdtOqqOPr0Te9fpuTWGA/bcx8V9AIGMLhgLsWwID2rz2odDPfBEtTccrzea8M9Hp5QtXq9EUDk+3k35DLNcwWJpR8dCy3Njn6S0BjKXjDrpxASBVGMSCPFaJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=e^{u}\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)" }, { "type": "step", "primary": "Substitute back $$u=x-y^{2}+xy$$", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAlUh/ymHcLfIOPgpzxU6cb6DBmrPPktdTvxtTacQJnvctRUNahunyblodcCh3llU2Uf3okNm7NMKpx8+U71HC/BOzGqmM0kKF1tC3aDrTtyJbjUaFLVySTwkjkiRwGQK1+dxvxkRJJpe/LYL9fVwZX9Z8+21SeDds3T/csmEBlNrD/0vCq4AUGXlvMnnmiYsMGqkIX6kseCKdEth+cILnwyMX9qYKk0viGvXKHsQSVFeoi+ZPlQ6KBam0kIORBN9aU=" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)=y+1$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)-\\frac{\\partial\\:}{\\partial\\:x}\\left(y^{2}\\right)+\\frac{\\partial\\:}{\\partial\\:x}\\left(xy\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmuHQGTAre0/umYO3/E+LF4lyEB4JYjIUjkjbDZ4tfSJ+yeROYBotscHIZETI6FSe7NWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CdhSH/V18j9Kf/3yKXdVwr8kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(y^{2}\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(y^{2}\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAm0G1L0XPtAdhL9J5beOlXLHI5S0StY1FdtOqqOPr0TewZLuDut1hvgJ5ZB48DbyVWjeh7+jKEzLb7VNCEMF3Z/ca9U3ttcLbKPPWP74hSSwM9h7hTO83qlGCf7HoriitB6ZQejymexKwHa0BQlABl3" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(xy\\right)=y$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(xy\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=y\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "result": "=y\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=y", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAkCHrPsxDHNAfe2TPFce/rinFjOV6V4e2DrBKqW1EhFu5KCf/kU7OLqVO3RHXJ9Wv0OG38IleojCyebAtZy+3Tm9JiwEB0ZXmaMqMWNNpbCrjG1uBlSxEBevFU4pf3SPbWwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=1-0+y" }, { "type": "step", "primary": "Simplify", "result": "=y+1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-e^{x-y^{2}+xy}\\left(y+1\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(-e^{x-y^{2}+xy}\\right)=-e^{x-y^{2}+xy}\\left(-2y+x\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(-e^{x-y^{2}+xy}\\right)", "steps": [ { "type": "step", "primary": "Treat $$x\\:$$as a constant" }, { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\right)", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=x-y^{2}+xy$$" ], "result": "=\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "input": "\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "result": "=e^{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnor/Dc3XNYBnT8IDMcV0N5HI5S0StY1FdtOqqOPr0Te9fpuTWGA/bcx8V9AIGMLhgLsWwID2rz2odDPfBEtTccrzea8M9Hp5QtXq9EUDk+3k35DLNcwWJpR8dCy3Njn6S0BjKXjDrpxASBVGMSCPFaJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=e^{u}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)" }, { "type": "step", "primary": "Substitute back $$u=x-y^{2}+xy$$", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnJ4nz3803j+0CKUyt5RRs1DBmrPPktdTvxtTacQJnvctRUNahunyblodcCh3llU2Uf3okNm7NMKpx8+U71HC/BOzGqmM0kKF1tC3aDrTtyJbjUaFLVySTwkjkiRwGQK1+dxvxkRJJpe/LYL9fVwZX9Ws7P+XzlWrtLx+I7MWqjVT/0vCq4AUGXlvMnnmiYsMGqkIX6kseCKdEth+cILnwyMX9qYKk0viGvXKHsQSVFeoi+ZPlQ6KBam0kIORBN9aU=" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)=-2y+x$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)-\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{2}\\right)+\\frac{\\partial\\:}{\\partial\\:y}\\left(xy\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmFdqcv3y2g0UFFU/kpFz1blyEB4JYjIUjkjbDZ4tfSJ3+y6gfQnMr2Alg7BrHl9PbNWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CaFMIbUnDJGgiKdFUyJv5owkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{2}\\right)=2y$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2y^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2y", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmJlzijqamNN9cX2Te5ihJ8HI5S0StY1FdtOqqOPr0Te3OdgKTuBQ7b7VDodvCI+RRkS3dlcCKpQTQcheuut7Mkm+hmRJA1ZPgdMDAPJn089uYCyhDjtUC8bpHuo8jw3JnXWAAl+Kl9LwXkvhtHLYkp" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(xy\\right)=x$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(xy\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=x\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)=1$$", "result": "=x\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAlSL9E7dKoAR9XAFM5xqmwCnFjOV6V4e2DrBKqW1EhFu4xSWAquVNPtoc274CycqX4OG38IleojCyebAtZy+3Tm9JiwEB0ZXmaMqMWNNpbCrrJfciFSknOagGsUT5AviymwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=0-2y+x" }, { "type": "step", "primary": "Simplify", "result": "=-2y+x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-e^{x-y^{2}+xy}\\left(-2y+x\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "∇f\\left(x,\\:y\\right)=\\left(-e^{x-y^{2}+xy}\\left(y+1\\right),\\:-e^{x-y^{2}+xy}\\left(-2y+x\\right)\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Solve $$\\left(-e^{x-y^{2}+xy}\\left(y+1\\right),\\:-e^{x-y^{2}+xy}\\left(-2y+x\\right)\\right)=\\left(0,\\:0\\right):{\\quad}\\left(-2,\\:-1\\right)$$", "input": "\\begin{bmatrix}-e^{x-y^{2}+xy}\\left(y+1\\right)=0\\\\-e^{x-y^{2}+xy}\\left(-2y+x\\right)=0\\end{bmatrix}", "steps": [ { "type": "interim", "title": "Isolate $$y\\:$$for $$-e^{x-y^{2}+xy}\\left(1+y\\right)=0:{\\quad}y=-1$$", "input": "-e^{x-y^{2}+xy}\\left(1+y\\right)=0", "steps": [ { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$", "result": "e^{x-y^{2}+xy}=0\\lor\\:1+y=0" }, { "type": "interim", "title": "Solve $$e^{x-y^{2}+xy}=0:{\\quad}$$No Solution for $$y\\in\\mathbb{R}$$", "input": "e^{x-y^{2}+xy}=0", "steps": [ { "type": "step", "primary": "$$a^{f\\left(y\\right)}$$ cannot be zero or negative for $$y\\in\\mathbb{R}$$", "result": "\\mathrm{No\\:Solution\\:for}\\:y\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$1+y=0:{\\quad}y=-1$$", "input": "1+y=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "1+y=0", "result": "y=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "1+y-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "y=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "y=-1" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Isolate Title 2Eq" } }, { "type": "step", "primary": "Plug the solutions $$y=-1$$ into $$-e^{x-y^{2}+xy}\\left(x-2y\\right)=0$$" }, { "type": "interim", "title": "For $$-e^{x-y^{2}+xy}\\left(x-2y\\right)=0$$, subsitute $$y$$ with $$-1:{\\quad}x=-2$$", "steps": [ { "type": "step", "primary": "For $$-e^{x-y^{2}+xy}\\left(x-2y\\right)=0$$, subsitute $$y$$ with $$-1$$", "result": "-e^{x-\\left(-1\\right)^{2}+x\\left(-1\\right)}\\left(x-2\\left(-1\\right)\\right)=0" }, { "type": "interim", "title": "Solve $$-e^{x-\\left(-1\\right)^{2}+x\\left(-1\\right)}\\left(x-2\\left(-1\\right)\\right)=0:{\\quad}x=-2$$", "input": "-e^{x-\\left(-1\\right)^{2}+x\\left(-1\\right)}\\left(x-2\\left(-1\\right)\\right)=0", "steps": [ { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$", "result": "e^{x-\\left(-1\\right)^{2}+x\\left(-1\\right)}=0\\lor\\:x-2\\left(-1\\right)=0" }, { "type": "interim", "title": "Solve $$e^{x-\\left(-1\\right)^{2}+x\\left(-1\\right)}=0:{\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "e^{x-\\left(-1\\right)^{2}+x\\left(-1\\right)}=0", "steps": [ { "type": "step", "primary": "$$a^{f\\left(x\\right)}$$ cannot be zero or negative for $$x\\in\\mathbb{R}$$", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$x-2\\left(-1\\right)=0:{\\quad}x=-2$$", "input": "x-2\\left(-1\\right)=0", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "x+2\\cdot\\:1=0" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "x+2=0" }, { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "x+2=0", "result": "x=-2", "steps": [ { "type": "step", "primary": "Subtract $$2$$ from both sides", "result": "x+2-2=0-2" }, { "type": "step", "primary": "Simplify", "result": "x=-2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=-2" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } } ], "meta": { "interimType": "Generic Substitute Specific 3Eq" } }, { "type": "step", "primary": "Plug the solutions $$x=-2$$ into $$-e^{x-y^{2}+xy}\\left(1+y\\right)=0$$" }, { "type": "interim", "title": "For $$-e^{x-y^{2}+xy}\\left(1+y\\right)=0$$, subsitute $$x$$ with $$-2:{\\quad}y=-1$$", "steps": [ { "type": "step", "primary": "For $$-e^{x-y^{2}+xy}\\left(1+y\\right)=0$$, subsitute $$x$$ with $$-2$$", "result": "-e^{-2-y^{2}+\\left(-2\\right)y}\\left(1+y\\right)=0" }, { "type": "interim", "title": "Solve $$-e^{-2-y^{2}+\\left(-2\\right)y}\\left(1+y\\right)=0:{\\quad}y=-1$$", "input": "-e^{-2-y^{2}+\\left(-2\\right)y}\\left(1+y\\right)=0", "steps": [ { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$", "result": "e^{-2-y^{2}+\\left(-2\\right)y}=0\\lor\\:1+y=0" }, { "type": "interim", "title": "Solve $$e^{-2-y^{2}+\\left(-2\\right)y}=0:{\\quad}$$No Solution for $$y\\in\\mathbb{R}$$", "input": "e^{-2-y^{2}+\\left(-2\\right)y}=0", "steps": [ { "type": "step", "primary": "$$a^{f\\left(y\\right)}$$ cannot be zero or negative for $$y\\in\\mathbb{R}$$", "result": "\\mathrm{No\\:Solution\\:for}\\:y\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$1+y=0:{\\quad}y=-1$$", "input": "1+y=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "1+y=0", "result": "y=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "1+y-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "y=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "y=-1" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } } ], "meta": { "interimType": "Generic Substitute Specific 3Eq" } }, { "type": "interim", "title": "Verify solutions by plugging them into the original equations", "steps": [ { "type": "step", "primary": "Check the solutions by plugging them into $$-e^{x-y^{2}+xy}\\left(y+1\\right)=0$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Check the solution $$x=-2,\\:y=-1:{\\quad}$$True", "input": "-e^{x-y^{2}+xy}\\left(y+1\\right)=0", "steps": [ { "type": "step", "primary": "Plug in $$x=-2,\\:y=-1$$", "result": "-e^{-2-\\left(-1\\right)^{2}+\\left(-2\\right)\\left(-1\\right)}\\left(-1+1\\right)=0" }, { "type": "step", "primary": "Refine", "result": "0=0" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Check One Solution 1Eq" } }, { "type": "step", "primary": "Check the solutions by plugging them into $$-e^{x-y^{2}+xy}\\left(-2y+x\\right)=0$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Check the solution $$x=-2,\\:y=-1:{\\quad}$$True", "input": "-e^{x-y^{2}+xy}\\left(-2y+x\\right)=0", "steps": [ { "type": "step", "primary": "Plug in $$x=-2,\\:y=-1$$", "result": "-e^{-2-\\left(-1\\right)^{2}+\\left(-2\\right)\\left(-1\\right)}\\left(-2\\left(-1\\right)-2\\right)=0" }, { "type": "step", "primary": "Refine", "result": "0=0" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Check One Solution 1Eq" } } ], "meta": { "interimType": "Check Solutions Plug Preface (many) 0Eq" } }, { "type": "step", "primary": "Therefore, the final solution for $$-e^{x-y^{2}+xy}\\left(y+1\\right)=0,\\:-e^{x-y^{2}+xy}\\left(-2y+x\\right)=0$$ is ", "result": "\\begin{pmatrix}x=-2,\\:&y=-1\\end{pmatrix}" } ], "meta": { "solvingClass": "System of Equations", "interimType": "Nonlinear Top 0Eq" } }, { "type": "step", "result": "\\left(-2,\\:-1\\right)" } ], "meta": { "interimType": "Explore Function Slope Zero 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7owSbZQBVuOtvRJx3LSI/x1x9GlaU51LokjxP5mluEl4oZr0lm52BdBZ6PiacdPVzPMdxWo2To9HCIiYIK+U1IjybDDQWyLHL1Zyj6inJGb8LIW9GykxoS93bIuVwW3dU8Jlxxus4QCiFdk6x0edSJtbA+zX4bD3u3gx65o2NJhMDkJhaD4X40KwIVm/QTZ8m/Xa8ghtocFM01zfh6gFLGibKyEHKUSAUXjCxpj7FLm1svqV0ZiqOJTpHnJ+sHuidvzIPeEtDfcHv/z8uls8Teg==" } } ], "meta": { "solvingClass": "Function Critical", "interimType": "Critical Points Table Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMs2QEmolszJqqcFPw9dQfeOEifYGdDNjOeEhpjTPpcVVOoHFsqato2ihhwZcZ9MCVuw38+Lb7jnCWH9be5n4PC+/nQcCyu1bTiLDnNbYrMBB0riGFjapWLWiCH6GlSJrW" } }, { "type": "interim", "title": "$$\\frac{\\partial^{2}f}{\\partial\\:x^{2}}=-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}$$", "input": "\\frac{\\partial^{2}}{\\partial\\:y^{2}}\\left(-e^{x-y^{2}+xy}\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(-e^{x-y^{2}+xy}\\right)=-e^{x-y^{2}+xy}\\left(-2y+x\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(-e^{x-y^{2}+xy}\\right)", "steps": [ { "type": "step", "primary": "Treat $$x\\:$$as a constant" }, { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\right)", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=x-y^{2}+xy$$" ], "result": "=\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "input": "\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "result": "=e^{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnor/Dc3XNYBnT8IDMcV0N5HI5S0StY1FdtOqqOPr0Te9fpuTWGA/bcx8V9AIGMLhgLsWwID2rz2odDPfBEtTccrzea8M9Hp5QtXq9EUDk+3k35DLNcwWJpR8dCy3Njn6S0BjKXjDrpxASBVGMSCPFaJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=e^{u}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)" }, { "type": "step", "primary": "Substitute back $$u=x-y^{2}+xy$$", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnJ4nz3803j+0CKUyt5RRs1DBmrPPktdTvxtTacQJnvctRUNahunyblodcCh3llU2Uf3okNm7NMKpx8+U71HC/BOzGqmM0kKF1tC3aDrTtyJbjUaFLVySTwkjkiRwGQK1+dxvxkRJJpe/LYL9fVwZX9Ws7P+XzlWrtLx+I7MWqjVT/0vCq4AUGXlvMnnmiYsMGqkIX6kseCKdEth+cILnwyMX9qYKk0viGvXKHsQSVFeoi+ZPlQ6KBam0kIORBN9aU=" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)=-2y+x$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)-\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{2}\\right)+\\frac{\\partial\\:}{\\partial\\:y}\\left(xy\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmFdqcv3y2g0UFFU/kpFz1blyEB4JYjIUjkjbDZ4tfSJ3+y6gfQnMr2Alg7BrHl9PbNWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CaFMIbUnDJGgiKdFUyJv5owkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{2}\\right)=2y$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2y^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2y", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmJlzijqamNN9cX2Te5ihJ8HI5S0StY1FdtOqqOPr0Te3OdgKTuBQ7b7VDodvCI+RRkS3dlcCKpQTQcheuut7Mkm+hmRJA1ZPgdMDAPJn089uYCyhDjtUC8bpHuo8jw3JnXWAAl+Kl9LwXkvhtHLYkp" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(xy\\right)=x$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(xy\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=x\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)=1$$", "result": "=x\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAlSL9E7dKoAR9XAFM5xqmwCnFjOV6V4e2DrBKqW1EhFu4xSWAquVNPtoc274CycqX4OG38IleojCyebAtZy+3Tm9JiwEB0ZXmaMqMWNNpbCrrJfciFSknOagGsUT5AviymwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=0-2y+x" }, { "type": "step", "primary": "Simplify", "result": "=-2y+x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-e^{x-y^{2}+xy}\\left(-2y+x\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{\\partial\\:}{\\partial\\:y}\\left(-e^{x-y^{2}+xy}\\left(-2y+x\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(-e^{x-y^{2}+xy}\\left(-2y+x\\right)\\right)=-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(-e^{x-y^{2}+xy}\\left(-2y+x\\right)\\right)", "steps": [ { "type": "step", "primary": "Treat $$x\\:$$as a constant" }, { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\left(-2y+x\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=e^{x-y^{2}+xy},\\:g=-2y+x$$" ], "result": "=-\\left(\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\right)\\left(-2y+x\\right)+\\frac{\\partial\\:}{\\partial\\:y}\\left(-2y+x\\right)e^{x-y^{2}+xy}\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\right)=e^{x-y^{2}+xy}\\left(-2y+x\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\right)", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=x-y^{2}+xy$$" ], "result": "=\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "input": "\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "result": "=e^{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnor/Dc3XNYBnT8IDMcV0N5HI5S0StY1FdtOqqOPr0Te9fpuTWGA/bcx8V9AIGMLhgLsWwID2rz2odDPfBEtTccrzea8M9Hp5QtXq9EUDk+3k35DLNcwWJpR8dCy3Njn6S0BjKXjDrpxASBVGMSCPFaJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=e^{u}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)" }, { "type": "step", "primary": "Substitute back $$u=x-y^{2}+xy$$", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnJ4nz3803j+0CKUyt5RRs1DBmrPPktdTvxtTacQJnvctRUNahunyblodcCh3llU2Uf3okNm7NMKpx8+U71HC/BOzGqmM0kKF1tC3aDrTtyJbjUaFLVySTwkjkiRwGQK1+dxvxkRJJpe/LYL9fVwZX9Ws7P+XzlWrtLx+I7MWqjVT/0vCq4AUGXlvMnnmiYsMGqkIX6kseCKdEth+cILnwyMX9qYKk0viGvXKHsQSVFeoi+ZPlQ6KBam0kIORBN9aU=" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)=-2y+x$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)-\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{2}\\right)+\\frac{\\partial\\:}{\\partial\\:y}\\left(xy\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmFdqcv3y2g0UFFU/kpFz1blyEB4JYjIUjkjbDZ4tfSJ3+y6gfQnMr2Alg7BrHl9PbNWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CaFMIbUnDJGgiKdFUyJv5owkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{2}\\right)=2y$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2y^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2y", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmJlzijqamNN9cX2Te5ihJ8HI5S0StY1FdtOqqOPr0Te3OdgKTuBQ7b7VDodvCI+RRkS3dlcCKpQTQcheuut7Mkm+hmRJA1ZPgdMDAPJn089uYCyhDjtUC8bpHuo8jw3JnXWAAl+Kl9LwXkvhtHLYkp" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(xy\\right)=x$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(xy\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=x\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)=1$$", "result": "=x\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAlSL9E7dKoAR9XAFM5xqmwCnFjOV6V4e2DrBKqW1EhFu4xSWAquVNPtoc274CycqX4OG38IleojCyebAtZy+3Tm9JiwEB0ZXmaMqMWNNpbCrrJfciFSknOagGsUT5AviymwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=0-2y+x" }, { "type": "step", "primary": "Simplify", "result": "=-2y+x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=e^{x-y^{2}+xy}\\left(-2y+x\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(-2y+x\\right)=-2$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(-2y+x\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{\\partial\\:}{\\partial\\:y}\\left(2y\\right)+\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(2y\\right)=2$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(2y\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnIOB1y/G07+/An2TUKIh3ynFjOV6V4e2DrBKqW1EhFu2EtmEvdU5EIfaZixikaOU0OG38IleojCyebAtZy+3Tm9JiwEB0ZXmaMqMWNNpbCrtrx4GHECwlUzuL+7hrEugOwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmFdqcv3y2g0UFFU/kpFz1blyEB4JYjIUjkjbDZ4tfSJ3+y6gfQnMr2Alg7BrHl9PbNWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CaFMIbUnDJGgiKdFUyJv5owkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=-2+0" }, { "type": "step", "primary": "Simplify", "result": "=-2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-\\left(e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(-2y+x\\right)+\\left(-2\\right)e^{x-y^{2}+xy}\\right)" }, { "type": "interim", "title": "Simplify $$-\\left(e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(-2y+x\\right)+\\left(-2\\right)e^{x-y^{2}+xy}\\right):{\\quad}-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}$$", "input": "-\\left(e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(-2y+x\\right)+\\left(-2\\right)e^{x-y^{2}+xy}\\right)", "result": "=-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\left(e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(-2y+x\\right)-2e^{x-y^{2}+xy}\\right)" }, { "type": "interim", "title": "$$e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(-2y+x\\right)=e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}$$", "input": "e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(-2y+x\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\left(-2y+x\\right)\\left(-2y+x\\right)=\\:\\left(-2y+x\\right)^{1+1}$$" ], "result": "=e^{x-y^{2}+xy}\\left(-2y+x\\right)^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7z80+Ae1b8NKZWsMUxluuFG4qojWe/KGK6Zk2BhkN3BFV00rpv8+ZC6TM10tVCSHs4rAUNmPx08LGpzIJSITbpKdvPHsJ+AEbdX63Ct573oqHRHiQfDUsCtnJ39gXNVhNUxwuTSX8588RFhlwg0fQVuHtQl+nb1jhqaAUpZ+tpSMdBj90NY6imMPLpgbqT0GuoU7IAATNWtCQbNhb9Z57ig==" } }, { "type": "step", "result": "=-\\left(e^{x+xy-y^{2}}\\left(x-2y\\right)^{2}-2e^{x+xy-y^{2}}\\right)" }, { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}\\right)-\\left(-2e^{x-y^{2}+xy}\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7it0nALM4EgN4OvACTxl4V+HtQl+nb1jhqaAUpZ+tpSN3Xry6wNYgYLS6d0F83Y0qadtRF0sKLkBYqlna/sY0vMzBWJotReR4P4m6RE6FZ2OFlZBbHD+Pwb1P8zOdiBcdu+Qnz+b1SIm6/DtP903z1FIU613wmZnKTnPApBwmUvKBBTEk/JQ2cZ9WKuRzClU7ogdNzxiPyyQo0Ci2cDZfp55flUBTo6vaqkRJUfSoyhDwArNhYYdRxeIueAPx/qfsborDoiX8GPa90/hHbZIuN7CI2sSeA74029n2yo277ZU=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$D\\left(x,\\:y\\right)=-e^{x-y^{2}+xy}\\left(y+1\\right)^{2}\\left(-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}\\right)-\\left(2e^{x-y^{2}+xy}y^{2}+2e^{x-y^{2}+xy}y-e^{x-y^{2}+xy}xy-e^{x-y^{2}+xy}x-e^{x-y^{2}+xy}\\right)^{2}$$", "input": "f\\left(x,\\:y\\right)=-e^{x-y^{2}+xy}", "steps": [ { "type": "step", "primary": "$$D\\left(x,\\:y\\right)=\\frac{\\partial^{2}f}{\\partial\\:x^{2}}\\frac{\\partial^{2}f}{\\partial\\:y^{2}}-\\left(\\frac{\\partial^{2}f}{\\partial\\:x\\partial\\:y}\\right)^{2}$$" }, { "type": "interim", "title": "$$\\frac{\\partial^{2}f}{\\partial\\:x^{2}}=-e^{x-y^{2}+xy}\\left(y+1\\right)^{2}$$", "input": "\\frac{\\partial^{2}}{\\partial\\:x^{2}}\\left(-e^{x-y^{2}+xy}\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(-e^{x-y^{2}+xy}\\right)=-e^{x-y^{2}+xy}\\left(y+1\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(-e^{x-y^{2}+xy}\\right)", "steps": [ { "type": "step", "primary": "Treat $$y\\:$$as a constant" }, { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-\\frac{\\partial\\:}{\\partial\\:x}\\left(e^{x-y^{2}+xy}\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(e^{x-y^{2}+xy}\\right)", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=x-y^{2}+xy$$" ], "result": "=\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "input": "\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "result": "=e^{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnor/Dc3XNYBnT8IDMcV0N5HI5S0StY1FdtOqqOPr0Te9fpuTWGA/bcx8V9AIGMLhgLsWwID2rz2odDPfBEtTccrzea8M9Hp5QtXq9EUDk+3k35DLNcwWJpR8dCy3Njn6S0BjKXjDrpxASBVGMSCPFaJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=e^{u}\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)" }, { "type": "step", "primary": "Substitute back $$u=x-y^{2}+xy$$", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAlUh/ymHcLfIOPgpzxU6cb6DBmrPPktdTvxtTacQJnvctRUNahunyblodcCh3llU2Uf3okNm7NMKpx8+U71HC/BOzGqmM0kKF1tC3aDrTtyJbjUaFLVySTwkjkiRwGQK1+dxvxkRJJpe/LYL9fVwZX9Z8+21SeDds3T/csmEBlNrD/0vCq4AUGXlvMnnmiYsMGqkIX6kseCKdEth+cILnwyMX9qYKk0viGvXKHsQSVFeoi+ZPlQ6KBam0kIORBN9aU=" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)=y+1$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)-\\frac{\\partial\\:}{\\partial\\:x}\\left(y^{2}\\right)+\\frac{\\partial\\:}{\\partial\\:x}\\left(xy\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmuHQGTAre0/umYO3/E+LF4lyEB4JYjIUjkjbDZ4tfSJ+yeROYBotscHIZETI6FSe7NWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CdhSH/V18j9Kf/3yKXdVwr8kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(y^{2}\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(y^{2}\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAm0G1L0XPtAdhL9J5beOlXLHI5S0StY1FdtOqqOPr0TewZLuDut1hvgJ5ZB48DbyVWjeh7+jKEzLb7VNCEMF3Z/ca9U3ttcLbKPPWP74hSSwM9h7hTO83qlGCf7HoriitB6ZQejymexKwHa0BQlABl3" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(xy\\right)=y$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(xy\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=y\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "result": "=y\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=y", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAkCHrPsxDHNAfe2TPFce/rinFjOV6V4e2DrBKqW1EhFu5KCf/kU7OLqVO3RHXJ9Wv0OG38IleojCyebAtZy+3Tm9JiwEB0ZXmaMqMWNNpbCrjG1uBlSxEBevFU4pf3SPbWwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=1-0+y" }, { "type": "step", "primary": "Simplify", "result": "=y+1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-e^{x-y^{2}+xy}\\left(y+1\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{\\partial\\:}{\\partial\\:x}\\left(-e^{x-y^{2}+xy}\\left(y+1\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(-e^{x-y^{2}+xy}\\left(y+1\\right)\\right)=-e^{x-y^{2}+xy}\\left(y+1\\right)^{2}$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(-e^{x-y^{2}+xy}\\left(y+1\\right)\\right)", "steps": [ { "type": "step", "primary": "Treat $$y\\:$$as a constant" }, { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-\\left(y+1\\right)\\frac{\\partial\\:}{\\partial\\:x}\\left(e^{x-y^{2}+xy}\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(e^{x-y^{2}+xy}\\right)", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=x-y^{2}+xy$$" ], "result": "=\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "input": "\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "result": "=e^{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnor/Dc3XNYBnT8IDMcV0N5HI5S0StY1FdtOqqOPr0Te9fpuTWGA/bcx8V9AIGMLhgLsWwID2rz2odDPfBEtTccrzea8M9Hp5QtXq9EUDk+3k35DLNcwWJpR8dCy3Njn6S0BjKXjDrpxASBVGMSCPFaJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=e^{u}\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)" }, { "type": "step", "primary": "Substitute back $$u=x-y^{2}+xy$$", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAlUh/ymHcLfIOPgpzxU6cb6DBmrPPktdTvxtTacQJnvctRUNahunyblodcCh3llU2Uf3okNm7NMKpx8+U71HC/BOzGqmM0kKF1tC3aDrTtyJbjUaFLVySTwkjkiRwGQK1+dxvxkRJJpe/LYL9fVwZX9Z8+21SeDds3T/csmEBlNrD/0vCq4AUGXlvMnnmiYsMGqkIX6kseCKdEth+cILnwyMX9qYKk0viGvXKHsQSVFeoi+ZPlQ6KBam0kIORBN9aU=" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)=y+1$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)-\\frac{\\partial\\:}{\\partial\\:x}\\left(y^{2}\\right)+\\frac{\\partial\\:}{\\partial\\:x}\\left(xy\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmuHQGTAre0/umYO3/E+LF4lyEB4JYjIUjkjbDZ4tfSJ+yeROYBotscHIZETI6FSe7NWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CdhSH/V18j9Kf/3yKXdVwr8kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(y^{2}\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(y^{2}\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAm0G1L0XPtAdhL9J5beOlXLHI5S0StY1FdtOqqOPr0TewZLuDut1hvgJ5ZB48DbyVWjeh7+jKEzLb7VNCEMF3Z/ca9U3ttcLbKPPWP74hSSwM9h7hTO83qlGCf7HoriitB6ZQejymexKwHa0BQlABl3" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(xy\\right)=y$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(xy\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=y\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "result": "=y\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=y", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAkCHrPsxDHNAfe2TPFce/rinFjOV6V4e2DrBKqW1EhFu5KCf/kU7OLqVO3RHXJ9Wv0OG38IleojCyebAtZy+3Tm9JiwEB0ZXmaMqMWNNpbCrjG1uBlSxEBevFU4pf3SPbWwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=1-0+y" }, { "type": "step", "primary": "Simplify", "result": "=y+1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-\\left(y+1\\right)e^{x-y^{2}+xy}\\left(y+1\\right)" }, { "type": "interim", "title": "Simplify $$-\\left(y+1\\right)e^{x-y^{2}+xy}\\left(y+1\\right):{\\quad}-e^{x-y^{2}+xy}\\left(y+1\\right)^{2}$$", "input": "-\\left(y+1\\right)e^{x-y^{2}+xy}\\left(y+1\\right)", "result": "=-e^{x-y^{2}+xy}\\left(y+1\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\left(y+1\\right)\\left(y+1\\right)=\\:\\left(y+1\\right)^{1+1}$$" ], "result": "=-e^{x-y^{2}+xy}\\left(y+1\\right)^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=-e^{x-y^{2}+xy}\\left(y+1\\right)^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s706qrJQ1Bd+WNXaRjpdxb9VDckBzvqWaE8oMGBU51Rdfdd47a0hQ8flDbGsI5To1d5CYw7rkY1fjSwDC4ofHM0r0K7s3CcTfbICrOsYsFO1w/y9DKGIPglJ+qMi9xDu2KaRI7GCp0HQz+zDw23axddGChDn6hVa+zxbWCnhN+mx12W7WnjeoUQ0pz9G54ulY7" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-e^{x-y^{2}+xy}\\left(y+1\\right)^{2}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{\\partial^{2}f}{\\partial\\:y^{2}}=-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}$$", "input": "\\frac{\\partial^{2}}{\\partial\\:y^{2}}\\left(-e^{x-y^{2}+xy}\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(-e^{x-y^{2}+xy}\\right)=-e^{x-y^{2}+xy}\\left(-2y+x\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(-e^{x-y^{2}+xy}\\right)", "steps": [ { "type": "step", "primary": "Treat $$x\\:$$as a constant" }, { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\right)", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=x-y^{2}+xy$$" ], "result": "=\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "input": "\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "result": "=e^{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnor/Dc3XNYBnT8IDMcV0N5HI5S0StY1FdtOqqOPr0Te9fpuTWGA/bcx8V9AIGMLhgLsWwID2rz2odDPfBEtTccrzea8M9Hp5QtXq9EUDk+3k35DLNcwWJpR8dCy3Njn6S0BjKXjDrpxASBVGMSCPFaJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=e^{u}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)" }, { "type": "step", "primary": "Substitute back $$u=x-y^{2}+xy$$", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnJ4nz3803j+0CKUyt5RRs1DBmrPPktdTvxtTacQJnvctRUNahunyblodcCh3llU2Uf3okNm7NMKpx8+U71HC/BOzGqmM0kKF1tC3aDrTtyJbjUaFLVySTwkjkiRwGQK1+dxvxkRJJpe/LYL9fVwZX9Ws7P+XzlWrtLx+I7MWqjVT/0vCq4AUGXlvMnnmiYsMGqkIX6kseCKdEth+cILnwyMX9qYKk0viGvXKHsQSVFeoi+ZPlQ6KBam0kIORBN9aU=" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)=-2y+x$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)-\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{2}\\right)+\\frac{\\partial\\:}{\\partial\\:y}\\left(xy\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmFdqcv3y2g0UFFU/kpFz1blyEB4JYjIUjkjbDZ4tfSJ3+y6gfQnMr2Alg7BrHl9PbNWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CaFMIbUnDJGgiKdFUyJv5owkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{2}\\right)=2y$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2y^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2y", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmJlzijqamNN9cX2Te5ihJ8HI5S0StY1FdtOqqOPr0Te3OdgKTuBQ7b7VDodvCI+RRkS3dlcCKpQTQcheuut7Mkm+hmRJA1ZPgdMDAPJn089uYCyhDjtUC8bpHuo8jw3JnXWAAl+Kl9LwXkvhtHLYkp" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(xy\\right)=x$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(xy\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=x\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)=1$$", "result": "=x\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAlSL9E7dKoAR9XAFM5xqmwCnFjOV6V4e2DrBKqW1EhFu4xSWAquVNPtoc274CycqX4OG38IleojCyebAtZy+3Tm9JiwEB0ZXmaMqMWNNpbCrrJfciFSknOagGsUT5AviymwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=0-2y+x" }, { "type": "step", "primary": "Simplify", "result": "=-2y+x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-e^{x-y^{2}+xy}\\left(-2y+x\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{\\partial\\:}{\\partial\\:y}\\left(-e^{x-y^{2}+xy}\\left(-2y+x\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(-e^{x-y^{2}+xy}\\left(-2y+x\\right)\\right)=-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(-e^{x-y^{2}+xy}\\left(-2y+x\\right)\\right)", "steps": [ { "type": "step", "primary": "Treat $$x\\:$$as a constant" }, { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\left(-2y+x\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=e^{x-y^{2}+xy},\\:g=-2y+x$$" ], "result": "=-\\left(\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\right)\\left(-2y+x\\right)+\\frac{\\partial\\:}{\\partial\\:y}\\left(-2y+x\\right)e^{x-y^{2}+xy}\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\right)=e^{x-y^{2}+xy}\\left(-2y+x\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\right)", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=x-y^{2}+xy$$" ], "result": "=\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "input": "\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "result": "=e^{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnor/Dc3XNYBnT8IDMcV0N5HI5S0StY1FdtOqqOPr0Te9fpuTWGA/bcx8V9AIGMLhgLsWwID2rz2odDPfBEtTccrzea8M9Hp5QtXq9EUDk+3k35DLNcwWJpR8dCy3Njn6S0BjKXjDrpxASBVGMSCPFaJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=e^{u}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)" }, { "type": "step", "primary": "Substitute back $$u=x-y^{2}+xy$$", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnJ4nz3803j+0CKUyt5RRs1DBmrPPktdTvxtTacQJnvctRUNahunyblodcCh3llU2Uf3okNm7NMKpx8+U71HC/BOzGqmM0kKF1tC3aDrTtyJbjUaFLVySTwkjkiRwGQK1+dxvxkRJJpe/LYL9fVwZX9Ws7P+XzlWrtLx+I7MWqjVT/0vCq4AUGXlvMnnmiYsMGqkIX6kseCKdEth+cILnwyMX9qYKk0viGvXKHsQSVFeoi+ZPlQ6KBam0kIORBN9aU=" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)=-2y+x$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)-\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{2}\\right)+\\frac{\\partial\\:}{\\partial\\:y}\\left(xy\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmFdqcv3y2g0UFFU/kpFz1blyEB4JYjIUjkjbDZ4tfSJ3+y6gfQnMr2Alg7BrHl9PbNWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CaFMIbUnDJGgiKdFUyJv5owkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{2}\\right)=2y$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2y^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2y", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmJlzijqamNN9cX2Te5ihJ8HI5S0StY1FdtOqqOPr0Te3OdgKTuBQ7b7VDodvCI+RRkS3dlcCKpQTQcheuut7Mkm+hmRJA1ZPgdMDAPJn089uYCyhDjtUC8bpHuo8jw3JnXWAAl+Kl9LwXkvhtHLYkp" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(xy\\right)=x$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(xy\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=x\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)=1$$", "result": "=x\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAlSL9E7dKoAR9XAFM5xqmwCnFjOV6V4e2DrBKqW1EhFu4xSWAquVNPtoc274CycqX4OG38IleojCyebAtZy+3Tm9JiwEB0ZXmaMqMWNNpbCrrJfciFSknOagGsUT5AviymwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=0-2y+x" }, { "type": "step", "primary": "Simplify", "result": "=-2y+x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=e^{x-y^{2}+xy}\\left(-2y+x\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(-2y+x\\right)=-2$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(-2y+x\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{\\partial\\:}{\\partial\\:y}\\left(2y\\right)+\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(2y\\right)=2$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(2y\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnIOB1y/G07+/An2TUKIh3ynFjOV6V4e2DrBKqW1EhFu2EtmEvdU5EIfaZixikaOU0OG38IleojCyebAtZy+3Tm9JiwEB0ZXmaMqMWNNpbCrtrx4GHECwlUzuL+7hrEugOwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmFdqcv3y2g0UFFU/kpFz1blyEB4JYjIUjkjbDZ4tfSJ3+y6gfQnMr2Alg7BrHl9PbNWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CaFMIbUnDJGgiKdFUyJv5owkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=-2+0" }, { "type": "step", "primary": "Simplify", "result": "=-2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-\\left(e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(-2y+x\\right)+\\left(-2\\right)e^{x-y^{2}+xy}\\right)" }, { "type": "interim", "title": "Simplify $$-\\left(e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(-2y+x\\right)+\\left(-2\\right)e^{x-y^{2}+xy}\\right):{\\quad}-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}$$", "input": "-\\left(e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(-2y+x\\right)+\\left(-2\\right)e^{x-y^{2}+xy}\\right)", "result": "=-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\left(e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(-2y+x\\right)-2e^{x-y^{2}+xy}\\right)" }, { "type": "interim", "title": "$$e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(-2y+x\\right)=e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}$$", "input": "e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(-2y+x\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\left(-2y+x\\right)\\left(-2y+x\\right)=\\:\\left(-2y+x\\right)^{1+1}$$" ], "result": "=e^{x-y^{2}+xy}\\left(-2y+x\\right)^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7z80+Ae1b8NKZWsMUxluuFG4qojWe/KGK6Zk2BhkN3BFV00rpv8+ZC6TM10tVCSHs4rAUNmPx08LGpzIJSITbpKdvPHsJ+AEbdX63Ct573oqHRHiQfDUsCtnJ39gXNVhNUxwuTSX8588RFhlwg0fQVuHtQl+nb1jhqaAUpZ+tpSMdBj90NY6imMPLpgbqT0GuoU7IAATNWtCQbNhb9Z57ig==" } }, { "type": "step", "result": "=-\\left(e^{x+xy-y^{2}}\\left(x-2y\\right)^{2}-2e^{x+xy-y^{2}}\\right)" }, { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}\\right)-\\left(-2e^{x-y^{2}+xy}\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7it0nALM4EgN4OvACTxl4V+HtQl+nb1jhqaAUpZ+tpSN3Xry6wNYgYLS6d0F83Y0qadtRF0sKLkBYqlna/sY0vMzBWJotReR4P4m6RE6FZ2OFlZBbHD+Pwb1P8zOdiBcdu+Qnz+b1SIm6/DtP903z1FIU613wmZnKTnPApBwmUvKBBTEk/JQ2cZ9WKuRzClU7ogdNzxiPyyQo0Ci2cDZfp55flUBTo6vaqkRJUfSoyhDwArNhYYdRxeIueAPx/qfsborDoiX8GPa90/hHbZIuN7CI2sSeA74029n2yo277ZU=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{\\partial^{2}f}{\\partial\\:x\\partial\\:y}=2e^{x-y^{2}+xy}y^{2}+2e^{x-y^{2}+xy}y-e^{x-y^{2}+xy}xy-e^{x-y^{2}+xy}x-e^{x-y^{2}+xy}$$", "input": "\\frac{\\partial^{2}}{\\partial\\:x\\partial\\:y}\\left(-e^{x-y^{2}+xy}\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(-e^{x-y^{2}+xy}\\right)=-e^{x-y^{2}+xy}\\left(y+1\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(-e^{x-y^{2}+xy}\\right)", "steps": [ { "type": "step", "primary": "Treat $$y\\:$$as a constant" }, { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-\\frac{\\partial\\:}{\\partial\\:x}\\left(e^{x-y^{2}+xy}\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(e^{x-y^{2}+xy}\\right)", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=x-y^{2}+xy$$" ], "result": "=\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "input": "\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "result": "=e^{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnor/Dc3XNYBnT8IDMcV0N5HI5S0StY1FdtOqqOPr0Te9fpuTWGA/bcx8V9AIGMLhgLsWwID2rz2odDPfBEtTccrzea8M9Hp5QtXq9EUDk+3k35DLNcwWJpR8dCy3Njn6S0BjKXjDrpxASBVGMSCPFaJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=e^{u}\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)" }, { "type": "step", "primary": "Substitute back $$u=x-y^{2}+xy$$", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAlUh/ymHcLfIOPgpzxU6cb6DBmrPPktdTvxtTacQJnvctRUNahunyblodcCh3llU2Uf3okNm7NMKpx8+U71HC/BOzGqmM0kKF1tC3aDrTtyJbjUaFLVySTwkjkiRwGQK1+dxvxkRJJpe/LYL9fVwZX9Z8+21SeDds3T/csmEBlNrD/0vCq4AUGXlvMnnmiYsMGqkIX6kseCKdEth+cILnwyMX9qYKk0viGvXKHsQSVFeoi+ZPlQ6KBam0kIORBN9aU=" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)=y+1$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)-\\frac{\\partial\\:}{\\partial\\:x}\\left(y^{2}\\right)+\\frac{\\partial\\:}{\\partial\\:x}\\left(xy\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmuHQGTAre0/umYO3/E+LF4lyEB4JYjIUjkjbDZ4tfSJ+yeROYBotscHIZETI6FSe7NWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CdhSH/V18j9Kf/3yKXdVwr8kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(y^{2}\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(y^{2}\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAm0G1L0XPtAdhL9J5beOlXLHI5S0StY1FdtOqqOPr0TewZLuDut1hvgJ5ZB48DbyVWjeh7+jKEzLb7VNCEMF3Z/ca9U3ttcLbKPPWP74hSSwM9h7hTO83qlGCf7HoriitB6ZQejymexKwHa0BQlABl3" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(xy\\right)=y$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(xy\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=y\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "result": "=y\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=y", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAkCHrPsxDHNAfe2TPFce/rinFjOV6V4e2DrBKqW1EhFu5KCf/kU7OLqVO3RHXJ9Wv0OG38IleojCyebAtZy+3Tm9JiwEB0ZXmaMqMWNNpbCrjG1uBlSxEBevFU4pf3SPbWwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=1-0+y" }, { "type": "step", "primary": "Simplify", "result": "=y+1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-e^{x-y^{2}+xy}\\left(y+1\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{\\partial\\:}{\\partial\\:y}\\left(-e^{x-y^{2}+xy}\\left(y+1\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(-e^{x-y^{2}+xy}\\left(y+1\\right)\\right)=2e^{x-y^{2}+xy}y^{2}+2e^{x-y^{2}+xy}y-e^{x-y^{2}+xy}xy-e^{x-y^{2}+xy}x-e^{x-y^{2}+xy}$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(-e^{x-y^{2}+xy}\\left(y+1\\right)\\right)", "steps": [ { "type": "step", "primary": "Treat $$x\\:$$as a constant" }, { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\left(y+1\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=e^{x-y^{2}+xy},\\:g=y+1$$" ], "result": "=-\\left(\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\right)\\left(y+1\\right)+\\frac{\\partial\\:}{\\partial\\:y}\\left(y+1\\right)e^{x-y^{2}+xy}\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\right)=e^{x-y^{2}+xy}\\left(-2y+x\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(e^{x-y^{2}+xy}\\right)", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=x-y^{2}+xy$$" ], "result": "=\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "input": "\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:u}\\left(e^{u}\\right)=e^{u}$$", "result": "=e^{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnor/Dc3XNYBnT8IDMcV0N5HI5S0StY1FdtOqqOPr0Te9fpuTWGA/bcx8V9AIGMLhgLsWwID2rz2odDPfBEtTccrzea8M9Hp5QtXq9EUDk+3k35DLNcwWJpR8dCy3Njn6S0BjKXjDrpxASBVGMSCPFaJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=e^{u}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)" }, { "type": "step", "primary": "Substitute back $$u=x-y^{2}+xy$$", "result": "=e^{x-y^{2}+xy}\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnJ4nz3803j+0CKUyt5RRs1DBmrPPktdTvxtTacQJnvctRUNahunyblodcCh3llU2Uf3okNm7NMKpx8+U71HC/BOzGqmM0kKF1tC3aDrTtyJbjUaFLVySTwkjkiRwGQK1+dxvxkRJJpe/LYL9fVwZX9Ws7P+XzlWrtLx+I7MWqjVT/0vCq4AUGXlvMnnmiYsMGqkIX6kseCKdEth+cILnwyMX9qYKk0viGvXKHsQSVFeoi+ZPlQ6KBam0kIORBN9aU=" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)=-2y+x$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(x-y^{2}+xy\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)-\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{2}\\right)+\\frac{\\partial\\:}{\\partial\\:y}\\left(xy\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmFdqcv3y2g0UFFU/kpFz1blyEB4JYjIUjkjbDZ4tfSJ3+y6gfQnMr2Alg7BrHl9PbNWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CaFMIbUnDJGgiKdFUyJv5owkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{2}\\right)=2y$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2y^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2y", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmJlzijqamNN9cX2Te5ihJ8HI5S0StY1FdtOqqOPr0Te3OdgKTuBQ7b7VDodvCI+RRkS3dlcCKpQTQcheuut7Mkm+hmRJA1ZPgdMDAPJn089uYCyhDjtUC8bpHuo8jw3JnXWAAl+Kl9LwXkvhtHLYkp" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(xy\\right)=x$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(xy\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=x\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)=1$$", "result": "=x\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAlSL9E7dKoAR9XAFM5xqmwCnFjOV6V4e2DrBKqW1EhFu4xSWAquVNPtoc274CycqX4OG38IleojCyebAtZy+3Tm9JiwEB0ZXmaMqMWNNpbCrrJfciFSknOagGsUT5AviymwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=0-2y+x" }, { "type": "step", "primary": "Simplify", "result": "=-2y+x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=e^{x-y^{2}+xy}\\left(-2y+x\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(y+1\\right)=1$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(y+1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)+\\frac{\\partial\\:}{\\partial\\:y}\\left(1\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)=1$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:y}\\left(y\\right)=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnL8LERjbWuBxyItajsejw0lyEB4JYjIUjkjbDZ4tfSJ+yeROYBotscHIZETI6FSe7NWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CV1pRM1TxXJHihLW/JqDRmUkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(1\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(1\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnQ8agpheQE9sPaKrNUKTj9lyEB4JYjIUjkjbDZ4tfSJ3+y6gfQnMr2Alg7BrHl9PbNWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CfKylKcvm6AFSeVjh5E88c8kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=1+0" }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-\\left(e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(y+1\\right)+1\\cdot\\:e^{x-y^{2}+xy}\\right)" }, { "type": "interim", "title": "Simplify $$-\\left(e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(y+1\\right)+1\\cdot\\:e^{x-y^{2}+xy}\\right):{\\quad}2e^{x-y^{2}+xy}y^{2}+2e^{x-y^{2}+xy}y-e^{x-y^{2}+xy}xy-e^{x-y^{2}+xy}x-e^{x-y^{2}+xy}$$", "input": "-\\left(e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(y+1\\right)+1\\cdot\\:e^{x-y^{2}+xy}\\right)", "result": "=2e^{x-y^{2}+xy}y^{2}+2e^{x-y^{2}+xy}y-e^{x-y^{2}+xy}xy-e^{x-y^{2}+xy}x-e^{x-y^{2}+xy}", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:e^{x-y^{2}+xy}=e^{x-y^{2}+xy}$$", "result": "=-\\left(e^{x+xy-y^{2}}\\left(x-2y\\right)\\left(y+1\\right)+e^{x+xy-y^{2}}\\right)" }, { "type": "interim", "title": "Expand $$-\\left(e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(y+1\\right)+e^{x-y^{2}+xy}\\right):{\\quad}-e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(y+1\\right)-e^{x-y^{2}+xy}$$", "result": "=-e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(y+1\\right)-e^{x-y^{2}+xy}", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(y+1\\right)\\right)-\\left(e^{x-y^{2}+xy}\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(y+1\\right)-e^{x-y^{2}+xy}" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq" } }, { "type": "interim", "title": "Expand $$-e^{x-y^{2}+xy}\\left(-2y+x\\right)\\left(y+1\\right):{\\quad}2e^{x-y^{2}+xy}y^{2}+2e^{x-y^{2}+xy}y-e^{x-y^{2}+xy}xy-e^{x-y^{2}+xy}x$$", "result": "=2e^{x-y^{2}+xy}y^{2}+2e^{x-y^{2}+xy}y-e^{x-y^{2}+xy}xy-e^{x-y^{2}+xy}x-e^{x-y^{2}+xy}", "steps": [ { "type": "interim", "title": "Expand $$\\left(-2y+x\\right)\\left(y+1\\right):{\\quad}-2y^{2}-2y+xy+x$$", "input": "\\left(-2y+x\\right)\\left(y+1\\right)", "result": "=-e^{x-y^{2}+xy}\\left(-2y^{2}-2y+xy+x\\right)", "steps": [ { "type": "step", "primary": "Apply FOIL method: $$\\left(a+b\\right)\\left(c+d\\right)=ac+ad+bc+bd$$", "secondary": [ "$$a=-2y,\\:b=x,\\:c=y,\\:d=1$$" ], "result": "=\\left(-2y\\right)y+\\left(-2y\\right)\\cdot\\:1+xy+x\\cdot\\:1", "meta": { "title": { "extension": "F-First<br/>O-Outer<br/>I-Inner<br/>L-Last" }, "practiceLink": "/practice/expansion-practice#area=main&subtopic=FOIL_Basic", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-2yy-2\\cdot\\:1\\cdot\\:y+xy+1\\cdot\\:x" }, { "type": "interim", "title": "Simplify $$-2yy-2\\cdot\\:1\\cdot\\:y+xy+1\\cdot\\:x:{\\quad}-2y^{2}-2y+xy+x$$", "input": "-2yy-2\\cdot\\:1\\cdot\\:y+xy+1\\cdot\\:x", "result": "=-2y^{2}-2y+xy+x", "steps": [ { "type": "interim", "title": "$$2yy=2y^{2}$$", "input": "2yy", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$yy=\\:y^{1+1}$$" ], "result": "=2y^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2y^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7N+f0q8/ax2DUfseKz+7Z9nCQoYlYQ8U+Tfyx0kyzI8gL/szAnEXS1SLQugePdlQPwK5wYJ43KTNvTpSxHb6rHo8BPOx0wlsgFN8qUa6AzA0=" } }, { "type": "interim", "title": "$$2\\cdot\\:1\\cdot\\:y=2y$$", "input": "2\\cdot\\:1\\cdot\\:y", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2y" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ht1jgkupea8+jUgT+M/VFH+ZCeZ1HdwAXplX78369OejkVi15I8rBefLi4Iyt2wrNRuuMe0Vv7e2hSloVfCOFXuMoyubzblmmWXBbGmpBNDLtjQjjUC3wymqP9760vvZ" } }, { "type": "interim", "title": "$$1\\cdot\\:x=x$$", "input": "1\\cdot\\:x", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:x=x$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ASx2YODBupsHY/9yrO15bd13jtrSFDx+UNsawjlOjV3pfPCe8nQAZY1bE89UDVgMPJrYhwc+zvuHrOLz58Ml2oD661lPR3w/W4zyCV9dwUw=" } }, { "type": "step", "result": "=-2y^{2}-2y+xy+x" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76kT9Iy4A70IwlR2VsDKdwVXTSum/z5kLpMzXS1UJIew0sLRtaXuUj3K9ZURMqmCZPNf9GnlsXsOT2QN2Fk9CZPC30sSftAIFS6Qkpy19IkqwKqDam34IoZycp2HoRH7eA/mo0W80psRB42ei/jlVXQ==" } }, { "type": "interim", "title": "Expand $$-e^{x-y^{2}+xy}\\left(-2y^{2}-2y+xy+x\\right):{\\quad}2e^{x-y^{2}+xy}y^{2}+2e^{x-y^{2}+xy}y-e^{x-y^{2}+xy}xy-e^{x-y^{2}+xy}x$$", "input": "-e^{x-y^{2}+xy}\\left(-2y^{2}-2y+xy+x\\right)", "result": "=2e^{x-y^{2}+xy}y^{2}+2e^{x-y^{2}+xy}y-e^{x-y^{2}+xy}xy-e^{x-y^{2}+xy}x", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=\\left(-e^{x-y^{2}+xy}\\right)\\left(-2y^{2}\\right)+\\left(-e^{x-y^{2}+xy}\\right)\\left(-2y\\right)+\\left(-e^{x-y^{2}+xy}\\right)xy+\\left(-e^{x-y^{2}+xy}\\right)x", "meta": { "title": { "extension": "Multiply each of the terms within the parentheses<br/>by the term outside the parenthesis" } } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$\\left(-a\\right)\\left(-b\\right)=ab,\\:\\:+\\left(-a\\right)=-a$$" ], "result": "=2e^{x-y^{2}+xy}y^{2}+2e^{x-y^{2}+xy}y-e^{x-y^{2}+xy}xy-e^{x-y^{2}+xy}x" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ap7opZ9WBnXERUL571I9mTlfBxYl/oq6TjnFwdLTk42O6Dd3zpuSnKXNDXc0BYg4s7BwSVdKEYSqn+4fKAQoGvpnVN8oOdXxTKbeRUJMYT7vWjevwZd0aub+/A8dka0y0hbMwOe/f6Pl4mR8TWLrbEXd5pWXIHc981lJGhOZTv2h8QTHtpiMVjDBh/DGodjYEnJ/zNrC2ARoMQpaswsTNvi4Vkk6HwHb3hqHiqPF/6GCkHw6kddKH++vdgfBI3W1Jl0NaNZH8JHwbdRq/M29GQ==" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7it0nALM4EgN4OvACTxl4Vw6PHJ83gqGC+EJLU1USK7qEKf4dHWNLLl+dfr4AuN74DngUj/SLAm3nlsDd2/ySg6uO77lnnV79llIPeBEfCMENCnSudGB4yMhGEwZ1On29uLCmX+A/SVUAGYSBVvWMpSITy7Sp68rgLRMjQi+rcJhKJLUmh53AKkP0bhXPfajtymSHNXUmaJM6v02E8Bfc4bvipsuQP4vIUKQe4xNI2Zl6pfF1z6umzUJTJvt+ojYZLR9ekOXDwag+OGfVyuetbVIPfXlnlxTBhSuliCVI8J8YqeHB0aZNWtg8yao/D3j4JtW1cEHj9L+vsBLviCB8CL8yD3hLQ33B7/8/LpbPE3o=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=2e^{x-y^{2}+xy}y^{2}+2e^{x-y^{2}+xy}y-e^{x-y^{2}+xy}xy-e^{x-y^{2}+xy}x-e^{x-y^{2}+xy}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "D\\left(x,\\:y\\right)=\\left(-e^{x-y^{2}+xy}\\left(y+1\\right)^{2}\\right)\\left(-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}\\right)-\\left(\\left(2e^{x-y^{2}+xy}y^{2}+2e^{x-y^{2}+xy}y-e^{x-y^{2}+xy}xy-e^{x-y^{2}+xy}x-e^{x-y^{2}+xy}\\right)\\right)^{2}" }, { "type": "interim", "title": "Simplify $$\\left(-e^{x-y^{2}+xy}\\left(y+1\\right)^{2}\\right)\\left(-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}\\right)-\\left(2e^{x-y^{2}+xy}y^{2}+2e^{x-y^{2}+xy}y-e^{x-y^{2}+xy}xy-e^{x-y^{2}+xy}x-e^{x-y^{2}+xy}\\right)^{2}:{\\quad}-e^{x-y^{2}+xy}\\left(y+1\\right)^{2}\\left(-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}\\right)-\\left(2e^{x-y^{2}+xy}y^{2}+2e^{x-y^{2}+xy}y-e^{x-y^{2}+xy}xy-e^{x-y^{2}+xy}x-e^{x-y^{2}+xy}\\right)^{2}$$", "input": "\\left(-e^{x-y^{2}+xy}\\left(y+1\\right)^{2}\\right)\\left(-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}\\right)-\\left(\\left(2e^{x-y^{2}+xy}y^{2}+2e^{x-y^{2}+xy}y-e^{x-y^{2}+xy}xy-e^{x-y^{2}+xy}x-e^{x-y^{2}+xy}\\right)\\right)^{2}", "result": "D\\left(x,\\:y\\right)=-e^{x-y^{2}+xy}\\left(y+1\\right)^{2}\\left(-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}\\right)-\\left(2e^{x-y^{2}+xy}y^{2}+2e^{x-y^{2}+xy}y-e^{x-y^{2}+xy}xy-e^{x-y^{2}+xy}x-e^{x-y^{2}+xy}\\right)^{2}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-e^{x-y^{2}+xy}\\left(y+1\\right)^{2}\\left(-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}\\right)-\\left(2e^{x-y^{2}+xy}y^{2}+2e^{x-y^{2}+xy}y-e^{x-y^{2}+xy}xy-e^{x-y^{2}+xy}x-e^{x-y^{2}+xy}\\right)^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "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" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Check the sign of at each critical point" }, { "type": "interim", "title": "Check critical point $$\\left(-2,\\:-1\\right):{\\quad}$$Saddle", "steps": [ { "type": "interim", "title": "Check the sign of $$D\\left(x,\\:y\\right)=-e^{x-y^{2}+xy}\\left(y+1\\right)^{2}\\left(-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}\\right)-\\left(2e^{x-y^{2}+xy}y^{2}+2e^{x-y^{2}+xy}y-e^{x-y^{2}+xy}xy-e^{x-y^{2}+xy}x-e^{x-y^{2}+xy}\\right)^{2}$$ at $$\\left(-2,\\:-1\\right):{\\quad}$$Negative", "steps": [ { "type": "step", "primary": "Plug $$\\left(-2,\\:-1\\right)\\:$$into $$-e^{x-y^{2}+xy}\\left(y+1\\right)^{2}\\left(-e^{x-y^{2}+xy}\\left(-2y+x\\right)^{2}+2e^{x-y^{2}+xy}\\right)-\\left(2e^{x-y^{2}+xy}y^{2}+2e^{x-y^{2}+xy}y-e^{x-y^{2}+xy}xy-e^{x-y^{2}+xy}x-e^{x-y^{2}+xy}\\right)^{2}$$", "result": "-e^{-2-\\left(-1\\right)^{2}+\\left(-2\\right)\\left(-1\\right)}\\left(-1+1\\right)^{2}\\left(-e^{-2-\\left(-1\\right)^{2}+\\left(-2\\right)\\left(-1\\right)}\\left(-2\\left(-1\\right)-2\\right)^{2}+2e^{-2-\\left(-1\\right)^{2}+\\left(-2\\right)\\left(-1\\right)}\\right)-\\left(2e^{-2-\\left(-1\\right)^{2}+\\left(-2\\right)\\left(-1\\right)}\\left(-1\\right)^{2}+2e^{-2-\\left(-1\\right)^{2}+\\left(-2\\right)\\left(-1\\right)}\\left(-1\\right)-e^{-2-\\left(-1\\right)^{2}+\\left(-2\\right)\\left(-1\\right)}\\left(-2\\right)\\left(-1\\right)-e^{-2-\\left(-1\\right)^{2}+\\left(-2\\right)\\left(-1\\right)}\\left(-2\\right)-e^{-2-\\left(-1\\right)^{2}+\\left(-2\\right)\\left(-1\\right)}\\right)^{2}" }, { "type": "step", "primary": "Simplify", "result": "-\\frac{1}{e^{2}}" }, { "type": "step", "result": "\\mathrm{Negative}" } ], "meta": { "interimType": "Check Positive Negative One Region 2Eq" } }, { "type": "step", "primary": "$$D\\left(x,\\:y\\right)<0,\\:$$therefore $$\\left(-2,\\:-1\\right)\\:$$is a saddle", "result": "\\mathrm{Saddle}\\left(-2,\\:-1\\right)" } ], "meta": { "interimType": "Check Critical Point Title 1Eq" } }, { "type": "step", "result": "\\mathrm{Saddle}\\left(-2,\\:-1\\right)" } ], "meta": { "solvingClass": "Function Extreme" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "-e^{x-y^{2}+xy}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }