derivative of y=ln(2x)

{ "query": { "display": "derivative of $$y=\\ln\\left(2x\\right)$$", "symbolab_question": "PRE_CALC#derivative y=\\ln(2x)" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\frac{1}{x}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\ln\\left(2x\\right)\\right)=\\frac{1}{x}$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(2x\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{1}{2x}\\frac{d}{dx}\\left(2x\\right)$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(2x\\right)\\right)", "result": "=\\frac{1}{2x}\\frac{d}{dx}\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\ln\\left(u\\right),\\:\\:u=2x$$" ], "result": "=\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)\\frac{d}{dx}\\left(2x\\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(2x\\right)" }, { "type": "step", "primary": "Substitute back $$u=2x$$", "result": "=\\frac{1}{2x}\\frac{d}{dx}\\left(2x\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjgc+L3QSQmb0rhR14S/trj8zeERICEnv1Ds5A1/BdIw2AW2hhsmfO/7M0RMNsJBcxhgKq+X573ilbTfYIBcNAzWM9HA0a5VHxu5rWRbGwYViSshf24Xf2esyeeKCdsFxiIim3Xa07T8DZYj24lqtW4EuDOVaQvKofqHoY5jNapszx1SRmkdwriKeTV96yww0ImpXFf3SOUx+H18qfp3MLg=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2x\\right)=2$$", "input": "\\frac{d}{dx}\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg2sQzwGEAAPyDk8n13Ps8XZGku9zFkxwe1dTH8vycb94wHsFp27x8BxzSfXYcuPllNbbqpyK7JQEZdATEJR51jH4j/fzMjnIhJwos1vPNWw" } }, { "type": "step", "result": "=\\frac{1}{2x}\\cdot\\:2" }, { "type": "interim", "title": "Simplify $$\\frac{1}{2x}\\cdot\\:2:{\\quad}\\frac{1}{x}$$", "input": "\\frac{1}{2x}\\cdot\\:2", "result": "=\\frac{1}{x}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2x}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{1}{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gLp7aMENR6+PTEWGJGMdkf4CsFtYCGVpnwQ6GR+/iwQJQJZuTAY5js+oqjdT8kslr478qsqVqgwLmYYu6zfToT/L0MoYg+CUn6oyL3EO7YppEjsYKnQdDP7MPDbdrF10CYG+FX0O9BTyh/mIPdKs67IYq4kzi+VLp/c/GhqKC9c=" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{1}{x}", "displayFormula": "y=\\frac{1}{x}", "derivativeFormula": "-\\frac{1}{x^{2}}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false }, "xmax": 0, "calculatePoints": true }, { "evalFormula": "y=\\frac{1}{x}", "displayFormula": "y=\\frac{1}{x}", "derivativeFormula": "-\\frac{1}{x^{2}}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false }, "xmin": 0, "calculatePoints": true }, { "evalFormula": "x=0", "displayFormula": "y=\\frac{1}{x}", "attributes": { "color": "PURPLE", "lineType": "DASH", "labels": [ "\\mathrm{Vertical\\:Asymptote}" ], "isAsymptote": true } }, { "evalFormula": "y=0", "displayFormula": "y=\\frac{1}{x}", "attributes": { "color": "PURPLE", "lineType": "DASH", "labels": [ 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