(\partial)/(\partial x)((x-1)/(x+1))

{ "query": { "display": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(\\frac{x-1}{x+1}\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{\\partial }{\\partial x}(\\frac{x-1}{x+1})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Partial Derivatives", "default": "\\frac{2}{(x+1)^{2}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(\\frac{x-1}{x+1}\\right)=\\frac{2}{\\left(x+1\\right)^{2}}$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(\\frac{x-1}{x+1}\\right)", "steps": [ { "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-1\\right)\\left(x+1\\right)-\\frac{\\partial\\:}{\\partial\\:x}\\left(x+1\\right)\\left(x-1\\right)}{\\left(x+1\\right)^{2}}" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(x-1\\right)=1$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(x-1\\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(1\\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": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(1\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\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+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAkJQloh76DUeoB3JfCRc1JYlyEB4JYjIUjkjbDZ4tfSJ3+y6gfQnMr2Alg7BrHl9PbNWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CTYJGKukFdo/SOyITpAF5ackt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=1-0" }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(x+1\\right)=1$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(x+1\\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(1\\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": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(1\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\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+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAkJQloh76DUeoB3JfCRc1JYlyEB4JYjIUjkjbDZ4tfSJ3+y6gfQnMr2Alg7BrHl9PbNWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CTYJGKukFdo/SOyITpAF5ackt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=1+0" }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1\\cdot\\:\\left(x+1\\right)-1\\cdot\\:\\left(x-1\\right)}{\\left(x+1\\right)^{2}}" }, { "type": "interim", "title": "Simplify $$\\frac{1\\cdot\\:\\left(x+1\\right)-1\\cdot\\:\\left(x-1\\right)}{\\left(x+1\\right)^{2}}:{\\quad}\\frac{2}{\\left(x+1\\right)^{2}}$$", "input": "\\frac{1\\cdot\\:\\left(x+1\\right)-1\\cdot\\:\\left(x-1\\right)}{\\left(x+1\\right)^{2}}", "result": "=\\frac{2}{\\left(x+1\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$1\\cdot\\:\\left(x+1\\right)-1\\cdot\\:\\left(x-1\\right)=x+1-\\left(x-1\\right)$$", "input": "1\\cdot\\:\\left(x+1\\right)-1\\cdot\\:\\left(x-1\\right)", "steps": [ { "type": "interim", "title": "$$1\\cdot\\:\\left(x+1\\right)=x+1$$", "input": "1\\cdot\\:\\left(x+1\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(x+1\\right)=\\left(x+1\\right)$$", "result": "=\\left(x+1\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=x+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78stcPSRCBBqcFrLI/SWHmC061ljBSPJeENOw2efoSWviICjW/kGDExjHB2UKz4pajFF+Grhte/2UqFkzsPs4p5PCz7mEBmYN3HVZu8uRWiiwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$1\\cdot\\:\\left(x-1\\right)=x-1$$", "input": "1\\cdot\\:\\left(x-1\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(x-1\\right)=\\left(x-1\\right)$$", "result": "=\\left(x-1\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7r4rQsjp+g8qQFhm3SL11QC061ljBSPJeENOw2efoSWuaJl8xJs1verjGFgUh5WIw7lvKtANBUJdQPS8f9+853OLVZC3Nkz+V3D6WzwTWdk7fX4J0nidMH+AfK8vr02z3" } }, { "type": "step", "result": "=x+1-\\left(x-1\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7IXePfxeMOgHRW1nB70PSdhfsbCn6KEvGUvq1OkK9Ia/ehkKrn0era9rz8TlL+x/vvmzFYMfBM9tFZbGFujfdxsgW5YRZGl3NZXlPOka+8ykeJhvjJ3qeKfnsL8QIBq3eUi+v14c8TadBrfAPAo5Fc6YDIrDD43l6ZVxsQfJbOb0=" } }, { "type": "step", "result": "=\\frac{x+1-\\left(x-1\\right)}{\\left(x+1\\right)^{2}}" }, { "type": "interim", "title": "Expand $$x+1-\\left(x-1\\right):{\\quad}2$$", "input": "x+1-\\left(x-1\\right)", "result": "=\\frac{2}{\\left(x+1\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$-\\left(x-1\\right):{\\quad}-x+1$$", "input": "-\\left(x-1\\right)", "result": "=x+1-x+1", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(x\\right)-\\left(-1\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-x+1" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Simplify $$x+1-x+1:{\\quad}2$$", "input": "x+1-x+1", "result": "=2", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=x-x+1+1" }, { "type": "step", "primary": "Add similar elements: $$x-x=0$$", "result": "=1+1" }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DK86lf7AW3U1DPFd1+mc1t13jtrSFDx+UNsawjlOjV3jAewWnbvHwHHNJ9dhy4+W1sD7NfhsPe7eDHrmjY0mEyjM+9V6xkbPT1DZ1S8X9q9julyH2s+TpD9oFPV/10Q0" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CLfSHLR0CGVpdHoWBOzGCjoc5x1XSCBgEKSOZEUsTJr+iar4r+a7HDk2u0E4O3JRAJYpRu9XpYrd8NSAW2DdD/KxLrO04AooUAReaJjhZCY4Uof5DhIqWFwEJxyWcjjg/z//r+dXk7h9vxeDCLuZqnKF3u2OIb4bFA3EO8aRlSWQq5yozHqYGypbegKEwhqYAFtAZAu8KlD6kp34t6k2BWDyUQoB/gBzENDzgP/RKbYWRUPDnIQXYMTDiaxDMpN7" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Partial%20Derivatives", "practiceTopic": "Partial Derivatives" } }, "meta": { "showVerify": true } }