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

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Chapter 3, 3.6 - Problem 47 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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embizze | High School Teacher | (Level 2) Educator Emeritus

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Differentiate `y=(cos(x))^x ` :

Take the natural logarithm of both sides:

`lny=ln(cos(x))^x `

Use the power property of logarithms:

`lny=xln(cos(x)) `

Differentiate; use the product rule on the RHS:

` (dy)/(dx)(1/y)=ln(cos(x))+x(-sin(x))/(cos(x)) `

`y'=y(ln(cos(x))-xtan(x)) `

Substituting for y we get:

` y'=(cos(x))^x(ln(cos(x))-xtan(x)) `

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