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

Textbook Question

Chapter 3, 3.6 - Problem 50 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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hkj1385 | (Level 1) Assistant Educator

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given: 

`y = (lnx)^cosx`  

taking log both sides we get

`lny = cosx*{ln(lnx)}`

Differentiating both sides we get

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

`or, dy/dx = y*[-sinx*{ln(lnx)} + {cosx/(xlnx)}`

`or, dy/dx = (lnx)^cosx[-sinx*{ln(lnx)} + {cosx/(x*lnx)}]`

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