Differentiation — Maths STD 12 Science — Question

Differentiating w.r.t. x, we get

$\begin{aligned} & \frac{d y}{d x}=\frac{d}{d x}\left[\cot \left(\frac{\log x}{2}\right)-\log \left(\frac{\cot x}{2}\right)\right] \\ & =\frac{d}{d x}\left[\cot \left(\frac{\log x}{2}\right)\right]-\frac{d}{d x}\left[\log \left(\frac{\cot x}{2}\right)\right] \\ & =-\operatorname{cosec}^2\left(\frac{\log x}{2}\right) \cdot \frac{d}{d x}\left(\frac{\log x}{2}\right)-\frac{1}{\left(\frac{\cot x}{2}\right)} \cdot \frac{d}{d x}\left(\frac{\cot x}{2}\right) \\ & =-\operatorname{cosec}^2\left(\frac{\log x}{2}\right) \times \frac{1}{2} \times \frac{1}{x}-\frac{2}{\cot x} \times \frac{1}{2} \times\left(-\operatorname{cosec}^2 x\right) \\ & =-\frac{\operatorname{cosec}^2\left(\frac{\log x}{2}\right)}{2 x}+\tan x \cdot \operatorname{cosec} 2 x\end{aligned}$