Using the chain rule to find derivative Could you help me figure this one out?
$$f(x) = 6e^{x\sin x}$$
$$f'(x) = \,?$$

Add logarithms to both sides of the equation $f(x) = 6e^{x \sin x}$ 
\begin{align}\ln f(x) &= \ln (6e^{x \sin x}) \\
&= \ln 6 + \ln e^{x \sin x} \\
&= \ln 6 + x \sin x,
\end{align}
and implicitly differentiate both sides to get
\begin{align}
\frac{f'(x)}{f(x)}=\sin x + x \cos x.
\end{align}
Finally, we get
\begin{align}
f'(x)&=f(x)(\sin x + x \cos x) \\ &= \boxed{6e^{x \sin x}(\sin x + x \cos x)}.
\end{align}   
