derivative of y=ln(sqrt(x))

{ "query": { "display": "derivative of $$y=\\ln\\left(\\sqrt{x}\\right)$$", "symbolab_question": "PRE_CALC#derivative y=\\ln(\\sqrt{x})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\frac{1}{2x}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\ln\\left(\\sqrt{x}\\right)\\right)=\\frac{1}{2x}$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(\\sqrt{x}\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{1}{\\sqrt{x}}\\frac{d}{dx}\\left(\\sqrt{x}\\right)$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(\\sqrt{x}\\right)\\right)", "result": "=\\frac{1}{\\sqrt{x}}\\frac{d}{dx}\\left(\\sqrt{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\ln\\left(u\\right),\\:\\:u=\\sqrt{x}$$" ], "result": "=\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)\\frac{d}{dx}\\left(\\sqrt{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(\\sqrt{x}\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sqrt{x}$$", "result": "=\\frac{1}{\\sqrt{x}}\\frac{d}{dx}\\left(\\sqrt{x}\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYrXiy6bUvhxwZcq3D6xEiUqaNohqAHcVbX8BTjr1ZRzoZ3GoG6Ko8jDPh4vymhs0+tlv8YVMwh/df5SMAfAmpJXv++bSprT8DRLjDQza+XRVtC0r1mGQza36BwJyqzDXt63CCMDvZs08hqQwvvEWeu4N5gj7/zD4e2EOslAgnMJAeqXxdc+rps1CUyb7fqI2Ga++0UXCFaDRjBhGCE8EEfmkQ/JnsM45+j77dquc6L38" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\sqrt{x}\\right)=\\frac{1}{2\\sqrt{x}}$$", "input": "\\frac{d}{dx}\\left(\\sqrt{x}\\right)", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\frac{d}{dx}\\left(x^{\\frac{1}{2}}\\right)", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=\\frac{1}{2}x^{\\frac{1}{2}-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}x^{\\frac{1}{2}-1}:{\\quad}\\frac{1}{2\\sqrt{x}}$$", "input": "\\frac{1}{2}x^{\\frac{1}{2}-1}", "result": "=\\frac{1}{2\\sqrt{x}}", "steps": [ { "type": "interim", "title": "$$x^{\\frac{1}{2}-1}=x^{-\\frac{1}{2}}$$", "input": "x^{\\frac{1}{2}-1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{1}{2}-1:{\\quad}-\\frac{1}{2}$$", "input": "\\frac{1}{2}-1", "result": "=x^{-\\frac{1}{2}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=-\\frac{1\\cdot\\:2}{2}+\\frac{1}{2}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-1\\cdot\\:2+1}{2}" }, { "type": "interim", "title": "$$-1\\cdot\\:2+1=-1$$", "input": "-1\\cdot\\:2+1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=-2+1" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-2+1=-1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s731snK5z/nd3Sq/6JpCqiX1XTSum/z5kLpMzXS1UJIew02FKSBoQo9V3G05AlnWtTyCE30rzMlUAIVDyhseMBropKGn5MuXZnb2ZCo/hVsBU=" } }, { "type": "step", "result": "=\\frac{-1}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{1}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qijEBDcyPMwV4Y1jeiGyoO0se7vRyav6BwUCJZptwG3MwViaLUXkeD+JukROhWdjQYCY06ctBCI/puUxKEtzAQH2kDe5DGYTz3TrPquGdIjtHZXPNLHlLyai31n5HH4G6M8osviUPEkWv33aMbZrSFQW3Chm7McvYpuS87Y5EFs=" } }, { "type": "step", "result": "=\\frac{1}{2}x^{-\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$x^{-\\frac{1}{2}}=\\frac{1}{\\sqrt{x}}$$" ], "result": "=\\frac{1}{2}\\cdot\\:\\frac{1}{\\sqrt{x}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:1}{2\\sqrt{x}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=\\frac{1}{2\\sqrt{x}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79noABpxc4IZFb3O0CFaPAbH6E/qPf7AlxQDX8MXU5OsAlilG71elit3w1IBbYN0P8rEus7TgCihQBF5omOFkJrT+HVv/GE/F+xidhei/cy8B9pA3uQxmE8906z6rhnSIHimBRYRqHSWeJkuUPhfTC1O468YRFxaQeTFqgRqR2ru/qNjapxCbBfMYIYTudnDYTk5AXTHU+C+TrGKWzqT97A==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{\\sqrt{x}}\\cdot\\:\\frac{1}{2\\sqrt{x}}" }, { "type": "interim", "title": "Simplify $$\\frac{1}{\\sqrt{x}}\\cdot\\:\\frac{1}{2\\sqrt{x}}:{\\quad}\\frac{1}{2x}$$", "input": "\\frac{1}{\\sqrt{x}}\\cdot\\:\\frac{1}{2\\sqrt{x}}", "result": "=\\frac{1}{2x}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:1}{\\sqrt{x}\\cdot\\:2\\sqrt{x}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=\\frac{1}{2\\sqrt{x}\\sqrt{x}}" }, { "type": "interim", "title": "$$\\sqrt{x}\\cdot\\:2\\sqrt{x}=2x$$", "input": "\\sqrt{x}\\cdot\\:2\\sqrt{x}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{x}\\sqrt{x}=x$$" ], "result": "=2x", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XHME8k6dYtmAcA7S9JNIChxo++zd0ae8hIUfb3X8IwMgJ/ZZA32ZInFBpDtxBfiKaT+s2p8IB/pLWZ6wXcUGUrtCR5dIjxQ5ASg+ZPFVSscgeVeJnmIGj4+nSp9O/ygS2uawQbqk/wx6oN0dOXv7Dw==" } }, { "type": "step", "result": "=\\frac{1}{2x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78B+yE0cCl8hT7YErNdnZGUxSW3EOahWkuvAUg0UpmnO0/h1b/xhPxfsYnYXov3MvRXYii9G6PtJS0VMzV5tSBP2i9gqKNBiEkMJvG7+cA4kYwL9tVmLoj6d0/fLN04EQ72wZm7kDUxdE6YSmfEbr2vKTjCE2IpPTfALSHOPVsNVLzllMZAV/GQHp8dEO4L7iXEErS/oLIoC+Y1JlHSxbUlNinSirXQuGVYWiKqz93SW/Mg94S0N9we//Py6WzxN6" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{1}{2x}", "displayFormula": "y=\\frac{1}{2x}", "derivativeFormula": "-\\frac{1}{2x^{2}}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false }, "xmax": 0, "calculatePoints": true }, { "evalFormula": "y=\\frac{1}{2x}", "displayFormula": "y=\\frac{1}{2x}", "derivativeFormula": "-\\frac{1}{2x^{2}}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false }, "xmin": 0, 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