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