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Question
If \[y = \log \sqrt{\tan x}, \text{ write } \frac{dy}{dx} \] ?
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Solution
\[\text{ We have, y } = \log\sqrt{\tan x}\]
\[ \Rightarrow y = \log \left( \tan x \right)^\frac{1}{2} \]
\[ \Rightarrow y = \frac{1}{2}\log \tan x \left[ \because \log a^b = b\log a \right]\]
\[\Rightarrow \frac{dy}{dx} = \frac{1}{2} \times \frac{1}{\tan x}\frac{d}{dx}\left( \tan x \right)\]
\[ \Rightarrow \frac{dy}{dx} = \frac{1}{2} \times \frac{1}{\tan x}\left( \sec^2 x \right)\]
\[ \Rightarrow \frac{dy}{dx} = \frac{1}{2\frac{\sin x}{\cos x} \times \cos^2 x}\]
\[ \Rightarrow \frac{dy}{dx} = \frac{1}{2 \sin x \cos x}\]
\[ \Rightarrow \frac{dy}{dx} = \frac{1}{\sin2x}\]
\[ \Rightarrow \frac{dy}{dx} = cosec \ 2x\]
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