Question

If $y={\cos }^{-1}\cos x,$ then $\frac{dy}{dx}$ at $x=\frac{5\pi }{4}$

• A. $1$
• B. $-1$
• C. $\frac{1}{\sqrt{2}}$
• D. $\frac{5\pi }{4}$

Answer 1

The correct option is B $-1$

Given,

$y={\cos }^{-1}\cos x$

$\frac{dy}{dx}$

$=\frac{d}{dx}\left({\cos }^{-1}\left(\cos \left(x\right)\right)\right)$

Let $f={\cos }^{-1}\left(u\right),u=\cos \left(x\right)$

$=\frac{d}{du}\left({\cos }^{-1}\left(u\right)\right)\frac{d}{dx}\left(\cos \left(x\right)\right)$

$=(-\frac{1}{\sqrt{1-u^2}})(-\sin \left(x\right))$

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

$\therefore \frac{dy}{dx}=\frac{\sin \left(x\right)}{\sqrt{{\sin }^2\left(x\right)}}$

${\frac{dy}{dx}}_{x=\frac{5\pi }{4}}=\frac{\sin \left(\frac{5\pi }{4}\right)}{\sqrt{{\sin }^2\left(\frac{5\pi }{4}\right)}}$

$=\frac{\sqrt{2}\sin \left(\frac{5\pi }{4}\right)}{1-\cos \left(\frac{\pi }{2}\right)}$

$=\frac{\sqrt{2}(-\frac{\sqrt{2}}{2})}{1-0}$

$=-1$