# Question #c9eb0

Mar 5, 2017

$6 x {\csc}^{2} \left(3 {x}^{2}\right)$

#### Explanation:

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder }} \textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{\frac{d}{\mathrm{dx}} \left(\cot x\right) = - {\csc}^{2} x} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

differentiate using the $\textcolor{b l u e}{\text{chain rule}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \times \frac{\mathrm{du}}{\mathrm{dx}}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{let } u = 3 {x}^{2} \Rightarrow \frac{\mathrm{du}}{\mathrm{dx}} = 6 x$

$\Rightarrow y = - \cot u \Rightarrow \frac{\mathrm{dy}}{\mathrm{du}} = - \left(- {\csc}^{2} u\right) = {\csc}^{2} u$

Complete the derivative, changing u back into terms of x

$\Rightarrow \frac{d}{\mathrm{dx}} \left(- \cot \left(3 {x}^{2}\right)\right) = {\csc}^{2} u \times 6 x = 6 x {\csc}^{2} \left(3 {x}^{2}\right)$