(\partial)/(\partial x)(x/(x^4-y^6))

{ "query": { "display": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(\\frac{x}{x^{4}-y^{6}}\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{\\partial }{\\partial x}(\\frac{x}{x^{4}-y^{6}})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Partial Derivatives", "default": "\\frac{-3x^{4}-y^{6}}{(x^{4}-y^{6})^{2}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(\\frac{x}{x^{4}-y^{6}}\\right)=\\frac{-3x^{4}-y^{6}}{\\left(x^{4}-y^{6}\\right)^{2}}$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(\\frac{x}{x^{4}-y^{6}}\\right)", "steps": [ { "type": "step", "primary": "Treat $$y\\:$$as a constant" }, { "type": "step", "primary": "Apply the Quotient Rule: $$\\left(\\frac{f}{g}\\right)'=\\frac{f'{\\cdot}g-g'{\\cdot}f}{g^{2}}$$", "result": "=\\frac{\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)\\left(x^{4}-y^{6}\\right)-\\frac{\\partial\\:}{\\partial\\:x}\\left(x^{4}-y^{6}\\right)x}{\\left(x^{4}-y^{6}\\right)^{2}}" }, { "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": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(x^{4}-y^{6}\\right)=4x^{3}$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(x^{4}-y^{6}\\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^{4}\\right)-\\frac{\\partial\\:}{\\partial\\:x}\\left(y^{6}\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(x^{4}\\right)=4x^{3}$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(x^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4x^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4x^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAk6DN0gJVnjP5s2jxjwzRflHI5S0StY1FdtOqqOPr0Te2P8dSlq21+AGrw8rCvQbXhmGJxy1X81oF4R6ivkdRhaMCpduEeI2njCEKkgMisPa5uBiDz/WIOhm2zsislrcVBy3r5UGqNsYgiYGxe+82ZhsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(y^{6}\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(y^{6}\\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+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAk+Nes2KeC50x+QLgi01yfjHI5S0StY1FdtOqqOPr0TewZLuDut1hvgJ5ZB48DbyVWjeh7+jKEzLb7VNCEMF3Z/ca9U3ttcLbKPPWP74hSSwM9h7hTO83qlGCf7HoriitAsYHhXwoqjzJY4wXDmBx/Z" } }, { "type": "step", "result": "=4x^{3}-0" }, { "type": "step", "primary": "Simplify", "result": "=4x^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1\\cdot\\:\\left(x^{4}-y^{6}\\right)-4x^{3}x}{\\left(x^{4}-y^{6}\\right)^{2}}" }, { "type": "interim", "title": "$$1\\cdot\\:\\left(x^{4}-y^{6}\\right)-4x^{3}x=-3x^{4}-y^{6}$$", "input": "1\\cdot\\:\\left(x^{4}-y^{6}\\right)-4x^{3}x", "result": "=\\frac{-3x^{4}-y^{6}}{\\left(x^{4}-y^{6}\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$1\\cdot\\:\\left(x^{4}-y^{6}\\right)=x^{4}-y^{6}$$", "input": "1\\cdot\\:\\left(x^{4}-y^{6}\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(x^{4}-y^{6}\\right)=\\left(x^{4}-y^{6}\\right)$$", "result": "=\\left(x^{4}-y^{6}\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=x^{4}-y^{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NzguR9mIOsTdHPSAL0T3j6SthMQvXw7MIpA46sJGII4JQJZuTAY5js+oqjdT8kslGYesqzdDNB+sMwwvJu9DrYxRfhq4bXv9lKhZM7D7OKdSlqN07Z9bZi60CVBQ8wEX1bOxSx+9eDPa1xiK1HwEAbCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$4x^{3}x=4x^{4}$$", "input": "4x^{3}x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$x^{3}x=\\:x^{3+1}$$" ], "result": "=4x^{3+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4x^{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EhdettwF34ShOipm550zSAOfOVs9mPIqDLV5QIWwt3nf2cR7HKBnAAMM/KQ+U9l2LwP2YqOynxvTKrslHAAgv+ea+zxLjA6GL1rZVqWf3XA=" } }, { "type": "step", "result": "=x^{4}-y^{6}-4x^{4}" }, { "type": "step", "primary": "Group like terms", "result": "=x^{4}-4x^{4}-y^{6}" }, { "type": "step", "primary": "Add similar elements: $$x^{4}-4x^{4}=-3x^{4}$$", "result": "=-3x^{4}-y^{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NzguR9mIOsTdHPSAL0T3jz+Brpd55FmW202x1VI16VctOtZYwUjyXhDTsNnn6ElrZyrtTO8tiqzgW3NPaiI4vuBkMdSA/4jWgFptHV2qjQ88mtiHBz7O+4es4vPnwyXatS8GRZ4yKN9E0d46nNPaIAelv9vSCUQdZT45t/gZcNQGISxffZgeYES2nawXWmYq" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Partial%20Derivatives", "practiceTopic": "Partial Derivatives" } }, "meta": { "showVerify": true } }