Question

If $y=\log \sqrt{\tan x}$, then the value of $\frac{dy}{dx}\mathrm{at}x=\frac{\pi }{4}$ is given by
(a) ∞
(b) 1
(c) 0
(d) $\frac{1}{2}$

Answer 1

(b) 1

$$\begin{gathered} \mathrm{We}\mathrm{have},y=\log \sqrt{\tan x} \\ \Rightarrow \frac{dy}{dx}=\frac{1}{\sqrt{\tan x}}\times \frac{d}{dx}\left(\sqrt{\tan x}\right) \\ \Rightarrow \frac{dy}{dx}=\frac{1}{\sqrt{\tan x}}\times \frac{1}{2\sqrt{\tan x}}\times \frac{d}{dx}\left(\tan x\right) \\ \Rightarrow \frac{dy}{dx}=\frac{{\sec }^{2}x}{2\tan x} \\ \mathrm{Now},{\left(\frac{dy}{dx}\right)}_{x=\frac{\mathrm{\pi }}{4}}=\frac{{\left[\sec \left(\frac{\mathrm{\pi }}{4}\right)\right]}^2}{2\tan \left({\displaystyle \frac{\mathrm{\pi }}{4}}\right)}=\frac{2}{2\times 1}=1 \end{gathered}$$