# Choose the correct alternative in the following:

Question:

Choose the correct alternative in the following:

If $\mathrm{y}=\log \sqrt{\tan \mathrm{x}}$, then the value of $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{x}=\frac{\pi}{4}$ is given by

A. $\infty$

B. 1

C. 0

D. $1 / 2$

Solution:

$\mathrm{y}=\log \sqrt{\tan \mathrm{x}}$

$\Rightarrow \mathrm{y}=\log (\tan \mathrm{x})^{\frac{1}{2}}$

$\Rightarrow \mathrm{y}=\frac{1}{2} \log (\tan \mathrm{x})$

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

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2} \cdot \frac{1}{\tan \mathrm{x}} \cdot\left(\sec ^{2} \mathrm{x}\right)$

$\Rightarrow\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{\mathrm{x}=\frac{\pi}{4}}=\frac{1}{2} \cdot \frac{1}{\tan \frac{\pi}{4}} \cdot\left(\sec ^{2} \frac{\pi}{4}\right)$

$\Rightarrow\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{\mathrm{x}=\frac{\pi}{4}}=\frac{1}{2} \cdot \frac{1}{1} \cdot(\sqrt{2})^{2}$

$\therefore\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{\mathrm{x}=\frac{\pi}{4}}=1$