# What is the derivative of x^(cosx)?

Jul 29, 2015

We shall attack this problem by logarithmic differentiation.

#### Explanation:

Let, $y = {x}^{\cos x}$

Taking natural logarithm on both the sides, we obtain

$L n y = C o s x L n x$

Now, differentiating with respect to $x$,

$\frac{1}{y} \frac{\mathrm{dy}}{\mathrm{dx}} = C o s \frac{x}{x} - S \in x L n x$

In the last step, I used chain rule on the LHS and the product rule on the RHS.

Thus, $\frac{\mathrm{dy}}{\mathrm{dx}} = {x}^{C} o s x \left[C o s \frac{x}{x} - S \in x L n x\right]$