Q:

Taking the derivative of $x^{\sin(e^x)}$

How am I suppose to take the derivative of $f(x)=x^{\sin(e^x)}$?
What should I make $u$ equals?
I tried to make $u=\sin(e^x)$ and $u=e^x$ but they didn't work.

A:

Hint: Here, you want to (start off) by using logarithmic differentiation - that is, take the derivative of $\ln f(x)$, then use it to find the derivative of $f$.
Note that
$$
\ln(x^{\sin(e^x)})=\sin(e^x)\ln(x).
$$
Try differentiating that.  Then, use 
$$
\frac{d}{dx}\left[\ln f(x)\right]=\frac{f'(x)}{f(x)}
$$
to solve for $f'(x)$.

