Question

If $y={\cos }^{–1}(\cos x)$, then $\frac{dy}{dx}$ is equal to ?

• A. $1$ in the whole plane
• B. $–1$ in the whole plane
• C. $1$ in the $2^{nd}$and $3^{rd}$ quadrants of the plane
• D. $–1$ in the $3^{rd}$ and $4^{th}$ quadrants of the plane

Answer 1

The correct option is D

$–1$ in the $3^{rd}$ and $4^{th}$ quadrants of the plane

Explanation for the correct option:

Find the value of $\frac{dy}{dx}$ :

Given,

$y={\cos }^{–1}(\cos x)$

Now, differentiate the given function with respect to $x$

$\begin{array}{rcl}\frac{dy}{dx}& =& \frac{-1}{\sqrt{(1-{\cos }^{2}x)}}\times (-\sin x)\mathbf{}\left[\because \frac{d\left({\cos }^{-1}x\right)}{\mathrm{dx}}=\frac{1}{\sqrt{1-x^2}},\frac{d}{\mathrm{dx}}\left(\mathrm{cosx}\right)=-\mathrm{sinx}\right]\\ & =& \frac{-1}{\sqrt{{\sin }^{2}x}}\times (-\sin x)\\ & =& \pm 1\end{array}$

Now find the range,

$$\begin{gathered} \frac{dy}{dx}=1forx\in [0,\pi ] \\ \frac{dy}{dx}=-1forx\in [\pi ,2\pi ] \end{gathered}$$

Hence, the correct option is D.