# How do you differentiate f(x)=cot(3x)  using the chain rule?

Feb 9, 2016

$\frac{\mathrm{du}}{\mathrm{dx}} = - 3 {\csc}^{2} \left(3 x\right)$

#### Explanation:

First, I assume you know that the derivative of $\cot x$ is $- {\csc}^{2} x$.

We substitute $u = 3 x$.

Therefore,

$\frac{\mathrm{du}}{\mathrm{dx}} = 3$.

Now we use the chain rule.

$\frac{d}{\mathrm{dx}} \left(\cot \left(3 x\right)\right) = \frac{d}{\mathrm{dx}} \left(\cot \left(u\right)\right)$

$= \frac{d}{\mathrm{du}} \left(\cot \left(u\right)\right) \cdot \frac{\mathrm{du}}{\mathrm{dx}}$

$= - {\csc}^{2} \left(u\right) \cdot 3$

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