`y = (sin(x))^ln(x)` Use logarithmic differentiation to find the derivative of the function.

Textbook Question

Chapter 3, 3.6 - Problem 48 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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hkj1385 | (Level 1) Assistant Educator

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given `y = (sinx)^(lnx)`

taking log to the base 'e' both sides we get,

`lny = lnx*(lnsinx)`

Differentiating both sides we get

`(1/y)*dy/dx = (1/x)*{lnsinx} + (cosx/sinx)*lnx`

or,

`y*dy/dx =(1/x)*{lnsinx} + (cotx)*lnx `

or, `dy/dx = y*[(1/x)*{lnsinx} + (cotx)*lnx]`

or, dy/dx = `(sinx)^(lnx)*[(1/x)*lnsinx + cotx*lnx]`

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