Question

If \[y = \log \sqrt{\tan x}\] then the value of \[\frac{dy}{dx}\text { at }x = \frac{\pi}{4}\] is given by __________ .

Options

• ∞

• 1

• 0

• `1/2`

Answer 1

`1`

 

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