Answer
$y^{\prime}(x)=-e^x \sin e^x -\sin x e^x +e^x \cos x$
Work Step by Step
We have: $y(x)= \cos (e^x)+e^x \cos x$
We differentiate both sides with respect to $x$.
$y^{\prime}(x)=\dfrac{d}{dx} [\cos (e^x)+e^x \cos x] \\= -\sin e^x \times e^x +e^x \times (-\sin x)+e^x \cos x$
Simplify to obtain:
$y^{\prime}(x)=-e^x \sin e^x -\sin x e^x +e^x \cos x$
