Question:
$\cos (a \cos x+b \sin x)$, for some constant $a$ and $b$.
Solution:
Let $y=\cos (a \cos x+b \sin x)$
By using chain rule, we obtain
$\frac{d y}{d x}=\frac{d}{d x} \cos (a \cos x+b \sin x)$
$\Rightarrow \frac{d y}{d x}=-\sin (a \cos x+b \sin x) \cdot \frac{d}{d x}(a \cos x+b \sin x)$
$=-\sin (a \cos x+b \sin x) \cdot[a(-\sin x)+b \cos x]$
$=(a \sin x-b \cos x) \cdot \sin (a \cos x+b \sin x)$