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

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Expert Answers

sciencesolve | Certified Educator

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)).` **