Answer
$$y' =-\sin x e^{\cos \left(x\right)} -e^x\sin \left(e^x\right)$$
Work Step by Step
Given
$$y= e^{cos\:x}+cos\left(e^x\right)$$
Then
\begin{align*}
y'&=\frac{d}{dx}\left(e^{\cos \left(x\right)}\right)+\frac{d}{dx}\left(\cos \left(e^x\right)\right)\\
&=-\sin x e^{\cos \left(x\right)} -e^x\sin \left(e^x\right)
\end{align*}
