derivative of ln(1/(1-x))

{ "query": { "display": "$$\\frac{d}{dx}\\left(\\ln\\left(\\frac{1}{1-x}\\right)\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{d}{dx}(\\ln(\\frac{1}{1-x}))" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\frac{1}{1-x}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\ln\\left(\\frac{1}{1-x}\\right)\\right)=\\frac{1}{1-x}$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(\\frac{1}{1-x}\\right)\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\ln\\left(\\frac{1}{1-x}\\right):{\\quad}-\\ln\\left(1-x\\right)$$", "input": "\\ln\\left(\\frac{1}{1-x}\\right)", "steps": [ { "type": "step", "primary": "Apply log rule: $$\\log_{a}\\left(\\frac{1}{x}\\right)=-\\log_{a}\\left(x\\right)$$", "result": "=-\\ln\\left(-x+1\\right)", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{d}{dx}\\left(-\\ln\\left(1-x\\right)\\right)" }, { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-\\frac{d}{dx}\\left(\\ln\\left(1-x\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{1}{1-x}\\frac{d}{dx}\\left(1-x\\right)$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(1-x\\right)\\right)", "result": "=\\frac{1}{1-x}\\frac{d}{dx}\\left(1-x\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\ln\\left(u\\right),\\:\\:u=1-x$$" ], "result": "=\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)\\frac{d}{dx}\\left(1-x\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)=\\frac{1}{u}$$", "input": "\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)=\\frac{1}{u}$$", "result": "=\\frac{1}{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYoqTCAmruKWcJsn66ZPDMT8cjlLRK1jUV206qo4+vRN78rEus7TgCihQBF5omOFkJq1PlbV5jLoKv9solFCc4blTW26qciuyUBGXQExCUedYd9mDo5FIvzrirtH7/W8pPUxk6YPA4jUd3Af4X0JJJ64=" } }, { "type": "step", "result": "=\\frac{1}{u}\\frac{d}{dx}\\left(1-x\\right)" }, { "type": "step", "primary": "Substitute back $$u=1-x$$", "result": "=\\frac{1}{1-x}\\frac{d}{dx}\\left(1-x\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnnZDzt2oZLUVmou50nACzqQp7tdIFyr1eVqMMLZHDTGOK1n91tyBoBr/ZHP0eNC/RSNU68ZmiYZN//Vg53tMEyDWCSqwfKB+LJy+pyziHhw6WgDs/9/Ow4Ia+3J0uMrXd4bBmgMCMR/fNG/lmu/V52qkIX6kseCKdEth+cILnwyMX9qYKk0viGvXKHsQSVFeoi+ZPlQ6KBam0kIORBN9aU=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(1-x\\right)=-1$$", "input": "\\frac{d}{dx}\\left(1-x\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(1\\right)-\\frac{dx}{dx}" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(1\\right)=0$$", "input": "\\frac{d}{dx}\\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+idP5vLVrjUll6eMdYmqQX14xoif/Hxcm4iYenIFJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTsKfyXa6Zj1lcQsTYejuhcz" } }, { "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": "step", "result": "=0-1" }, { "type": "step", "primary": "Simplify", "result": "=-1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-\\frac{1}{1-x}\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{1-x}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{1}{1-x}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }