Question

$\frac{d}{dx}\cot \left(\frac{x}{2}\right)=$

Answer 1

$\frac{d}{dx}\left(\cot x/2\right)=\frac{d}{dx}\left(\frac{\cos x/2}{\sin x/2}\right)$

$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$ (formula)

$=-\frac{1/2{\cos }^2\left(x/2\right)-\frac{1}{2}{\sin }^2\left(x/2\right)}{{\sin }^2\left(x/2\right)}$

$=\frac{-1/2}{{\sin }^2\left(x/2\right)}$

$=-\frac{1}{2}cose{c}^2\left(\frac{x}{2}\right)$.