Question

If $y=\log \sqrt{\tan x},\mathrm{write}\frac{dy}{dx}.$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have},y=\log \sqrt{\tan x} \\ \Rightarrow y=\log {\left(\tan x\right)}^{\frac{1}{2}} \\ \Rightarrow y=\frac{1}{2}\log \tan x\left[\because \log a^b=b\log a\right] \end{gathered}$$

$$\begin{gathered} \Rightarrow \frac{dy}{dx}=\frac{1}{2}\times \frac{1}{\tan x}\frac{d}{dx}\left(\tan x\right) \\ \Rightarrow \frac{dy}{dx}=\frac{1}{2}\times \frac{1}{\tan x}\left({\sec }^{2}x\right) \\ \Rightarrow \frac{dy}{dx}=\frac{1}{2{\displaystyle \frac{\sin x}{\cos x}}\times {\cos }^{2}x} \\ \Rightarrow \frac{dy}{dx}=\frac{1}{2\sin x\cos x} \\ \Rightarrow \frac{dy}{dx}=\frac{1}{\sin 2x} \\ \Rightarrow \frac{dy}{dx}=\mathrm{cosec}2x \end{gathered}$$