Question

Find $\frac{dy}{dx}$, if $y={\cos }^{-1}\left(\sin 5x\right)$

Answer 1

$y={\cos }^{-1}\left(\sin 5x\right)$

$\frac{dy}{dx}=\frac{-1}{\sqrt{1-\left({\sin }^{2}5x\right)}}\times \left(5\cos 5x\right)$

$=\frac{-5\cos 5x}{\left|\cos 5x\right|}$

$\therefore \frac{dy}{dx}=\{\begin{array}{c}-5x=\frac{2k\pi \pm \theta }{5},\\ 5x=\frac{(2k+1)\pi \pm \theta }{5}\end{array}$

$\theta \in [0,\frac{\pi }{2}]$