Question

Differentiate the function $\cos (a\cos x+b\sin x)$ w.r.t. $x$ for some constants $a$ and $b.$

Answer 1

Let $y=\cos (a\cos x+b\sin x)$

Differentiating w.r.t. $x$, we get

$\frac{dy}{dx}=\frac{d\left(\cos \right(a\cos x+b\sin x\left)\right)}{dx}$

$\frac{dy}{dx}=-\sin (a\cos x+b\sin x)\times \frac{d(a\cos x+b\sin x)}{dx}$

$(\text{Using chain rule}\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx})$

$\frac{dy}{dx}=-\sin (a\cos x+b\sin x)\times (a.\frac{d\left(\cos x\right)}{dx}+b.\frac{d\left(\sin x\right)}{dx})$

$\frac{dy}{dx}=\sin (a\cos x+b\sin x)\times (a\sin x-b\cos x)$