derivative of f(x)=ln(x^2)

{ "query": { "display": "derivative of $$f\\left(x\\right)=\\ln\\left(x^{2}\\right)$$", "symbolab_question": "PRE_CALC#derivative f(x)=\\ln(x^{2})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\frac{2}{x}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\ln\\left(x^{2}\\right)\\right)=\\frac{2}{x}$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(x^{2}\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{1}{x^{2}}\\frac{d}{dx}\\left(x^{2}\\right)$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(x^{2}\\right)\\right)", "result": "=\\frac{1}{x^{2}}\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\ln\\left(u\\right),\\:\\:u=x^{2}$$" ], "result": "=\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)\\frac{d}{dx}\\left(x^{2}\\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(x^{2}\\right)" }, { "type": "step", "primary": "Substitute back $$u=x^{2}$$", "result": "=\\frac{1}{x^{2}}\\frac{d}{dx}\\left(x^{2}\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg1+YUPeKHkBm0QbZbSXH5L7uWIGxmkxyDI+OHJC7dgBdLl7DeVd7l7l/uUT/v1GhNJLZOXSTCXkMFKW90A2Pi/484O9NJ/PDg7/ATYMA16bCbOYXAJO1XGOqdMzaDwAx3L6FnBN7fR1mVYFUJJgLIbvbBmbuQNTF0TphKZ8RuvabkCo9FlyJUsOgE5aAtC5rBG0MJHrQfXzSB22kHRiQ+E=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "step", "result": "=\\frac{1}{x^{2}}\\cdot\\:2x" }, { "type": "interim", "title": "Simplify $$\\frac{1}{x^{2}}\\cdot\\:2x:{\\quad}\\frac{2}{x}$$", "input": "\\frac{1}{x^{2}}\\cdot\\:2x", "result": "=\\frac{2}{x}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2x}{x^{2}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=\\frac{2x}{x^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=\\frac{2}{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7u2xELInXnjziDZRQn01yjhXIB5DLRSxA7lro+PXw7Hzdd47a0hQ8flDbGsI5To1doX/MZ3J6bCyfsTlPrpWwVxSB3l99+C+zGvPER86TQtCY3ASC+aZqPN1DBWUUsybFLOoCuoLwpnU8v8HDSwilTJ3dogIgQ1r9snaK0J17WEqwiNrEngO+NNvZ9sqNu+2V" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{2}{x}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }