Question

Find the derivative of $\text{cosec}x\cot x$.

Answer 1

Let $f\left(x\right)=\text{cosec}x\cot x$
Differentiating with respect to $x$
$\Rightarrow f^{\prime }\left(x\right)=\frac{d}{dx}\left(\text{cosec}x\cot x\right)$
$\Rightarrow f^{\prime }\left(x\right)=\left(\cot x\right)\frac{d}{dx}\left(\text{cosec}x\right)+\left(\text{cosec}x\right)\frac{d}{dx}\left(\cot x\right)$
$\Rightarrow f^{\prime }\left(x\right)=\left(\cot x\right)(-\text{cosec}x\cot x)+\left(\text{cosec}x\right)(-{\text{cosec}}^{2}x)$
$\therefore f^{\prime }\left(x\right)=-\text{cosec}x{\cot }^{2}x-{\text{cosec}}^{3}x$