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

Textbook Question

Chapter 3, 3.6 - Problem 45 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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Because the variable is raised to a variable power, you need to apply the natural logarithm both sides, such that:

`ln y = ln (x^(sin x))`

Using the property of logarithms, yields:

`ln y = sin x*ln x`

You need to differentiate both sides, using the chain rule to the left side, since y is a function of x, such that:

`(1/y)*y' = cos x*ln x + sin x*(1/x)`

`y' = y*(cos x*ln x + sin x*(1/x))`

Replacing `x^(sin x)` for y, yields:

`y' = (x^(sin x))*(cos x*ln x + sin x*(1/x))`

Hence, evaluating the derivative of the function, using logarithmic differentiation, yields `y' = (x^(sin x))*(cos x*ln x + sin x*(1/x)).`

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