What is the derivative of (cot^(2)x)?

Jan 9, 2016

$\frac{d}{\mathrm{dx}} \left({\cot}^{2} x\right) = \frac{- 2 \cos x}{{\sin}^{3} x}$

Explanation:

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

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

$= 2 \cot x \cdot \frac{\sin x \frac{d}{\mathrm{dx}} \left(\cos x\right) - \cos x \frac{d}{\mathrm{dx}} \left(\sin x\right)}{{\sin}^{2} x}$

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

$= 2 \cot x \cdot \frac{- \left({\sin}^{2} x + {\cos}^{2} x\right)}{{\sin}^{2} x}$

$= 2 \frac{\cos x}{\sin x} \cdot \frac{- 1}{{\sin}^{2} x}$

$= \frac{- 2 \cos x}{{\sin}^{3} x}$