Question

How to find the derivative of ${\mathrm{cosec}}^2\mathrm{x}$?

Answer 1

Calculation of derivative:

Formula: $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{cosec}\right(\mathrm{x}\left)\right)=-\mathrm{cosec}\left(\mathrm{x}\right)\cot \left(\mathrm{x}\right)$

Chain rule: $\frac{\mathrm{d}}{\mathrm{dx}}{\left[\mathrm{f}\left(\mathrm{x}\right)\right]}^2=2\mathrm{f}\left(\mathrm{x}\right)\frac{\mathrm{d}}{\mathrm{dx}}\mathrm{f}\left(\mathrm{x}\right)$

That is

$$\begin{gathered} \Rightarrow \frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{cosec}}^2\mathrm{x}\right)=2\mathrm{cosec}\left(\mathrm{x}\right)\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{cosec}\right(\mathrm{x}\left)\right)\mathbf{}[\because \mathrm{Chain}\mathrm{rule}] \\ \Rightarrow \frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{cosec}}^2\mathrm{x}\right)=2\mathrm{cosec}\left(\mathrm{x}\right)(-\mathrm{cosec}(\mathrm{x}\left)\cot \right(\mathrm{x}\left)\right)[\because \mathrm{Above}\mathrm{formula}] \\ \Rightarrow \frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{cosec}}^2\mathrm{x}\right)=-2{\mathrm{cosec}}^2\left(\mathrm{x}\right)\cot \left(\mathrm{x}\right) \end{gathered}$$

Therefore, the derivative of ${\mathrm{cosec}}^2\mathrm{x}=-2{\mathrm{cosec}}^2\left(\mathrm{x}\right)\cot \left(\mathrm{x}\right)$.