Question

Differentiate the following w.r.t. $x$ :
${\cos }^{-1}\left(\frac{3\cos x-4\sin x}{5}\right)$

Answer 1

$\frac{3\cos x-4\sin x}{5}=\frac{3\cos x}{5}-\frac{4\sin x}{5}=\cos 53°\cos x-\sin 53°\sin x=\cos (x-53°)$

$\frac{d}{dx}\left({\cos }^{-1}\frac{3\cos x-4\sin x}{5}\right)=\frac{d}{dx}{\cos }^{-1}\cos (x-53°)=\frac{d}{dx}(x-53°)=1$