derivative of f(x)= x/(x-2)

{ "query": { "display": "derivative of $$f\\left(x\\right)=\\frac{x}{x-2}$$", "symbolab_question": "PRE_CALC#derivative f(x)=\\frac{x}{x-2}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "-\\frac{2}{(x-2)^{2}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{x}{x-2}\\right)=-\\frac{2}{\\left(x-2\\right)^{2}}$$", "input": "\\frac{d}{dx}\\left(\\frac{x}{x-2}\\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{dx}{dx}\\left(x-2\\right)-\\frac{d}{dx}\\left(x-2\\right)x}{\\left(x-2\\right)^{2}}" }, { "type": "interim", "title": "$$\\frac{dx}{dx}=1$$", "input": "\\frac{dx}{dx}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYko/29fz701XcRtz4b42RqRjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIdz2XHFZ6BxfaHSMA6lT+lbVmoiKRd+ttkZ9NIrGodT+" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x-2\\right)=1$$", "input": "\\frac{d}{dx}\\left(x-2\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{dx}{dx}-\\frac{d}{dx}\\left(2\\right)" }, { "type": "interim", "title": "$$\\frac{dx}{dx}=1$$", "input": "\\frac{dx}{dx}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYko/29fz701XcRtz4b42RqRjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIdz2XHFZ6BxfaHSMA6lT+lbVmoiKRd+ttkZ9NIrGodT+" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2\\right)=0$$", "input": "\\frac{d}{dx}\\left(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+idP5vLVrjUll6eMdYiiraNd5UTAiEFXslV0UVyVJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTtRm0l+ci6m9OnlYfI6EjHe" } }, { "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-2\\right)-1\\cdot\\:x}{\\left(x-2\\right)^{2}}" }, { "type": "interim", "title": "Simplify $$\\frac{1\\cdot\\:\\left(x-2\\right)-1\\cdot\\:x}{\\left(x-2\\right)^{2}}:{\\quad}-\\frac{2}{\\left(x-2\\right)^{2}}$$", "input": "\\frac{1\\cdot\\:\\left(x-2\\right)-1\\cdot\\:x}{\\left(x-2\\right)^{2}}", "result": "=-\\frac{2}{\\left(x-2\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$1\\cdot\\:\\left(x-2\\right)-1\\cdot\\:x=-2$$", "input": "1\\cdot\\:\\left(x-2\\right)-1\\cdot\\:x", "steps": [ { "type": "interim", "title": "$$1\\cdot\\:\\left(x-2\\right)=x-2$$", "input": "1\\cdot\\:\\left(x-2\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(x-2\\right)=\\left(x-2\\right)$$", "result": "=\\left(x-2\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=x-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7w9zw2/yPFhISum1SDJ8FEC061ljBSPJeENOw2efoSWsnf/yYLVgeaYqBa9/NwR28jFF+Grhte/2UqFkzsPs4p+9vGKNPsQEnxhzsz2vphmiwiNrEngO+NNvZ9sqNu+2V" } }, { "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": "=x-2-x" }, { "type": "step", "primary": "Group like terms", "result": "=x-x-2" }, { "type": "step", "primary": "Add similar elements: $$x-x=0$$", "result": "=-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FR7DkJJoupByyoYqsArTu2MDvsCybD4clUUwNttKxcUDnzlbPZjyKgy1eUCFsLd5RDtWG5guT8Xmwx3nth5SgkvlJSNjJhmREx0p+VXCgxfKW4jfGzQ0lZN84R8yuds6X+5PwrYxdnGvuKcIJPZZhA==" } }, { "type": "step", "result": "=\\frac{-2}{\\left(x-2\\right)^{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{2}{\\left(x-2\\right)^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CLfSHLR0CGVpdHoWBOzGChSt20AEsCrGOir84+QERKOu8T7X8wm3WlnN5iWuQtJbICf2WQN9mSJxQaQ7cQX4iplokxKUI2rd0ju/5dpufJeFJ7Y1Kxil7fM5Cb5h/7vdo3oe/oyhMy2+1TQhDBd2f2zM6E3fuZxF1XkKAYaRXCAwbaQzbBJ4qVmNWRd0hBkK7bcDQLXbZSUhSo3wEc+BayFlGuKmmm2BlQWsSVb6hTA=" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=-\\frac{2}{(x-2)^{2}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }