# How do you differentiate  f(x)=cot(x)cos(x) using the product rule?

Oct 28, 2015

This is more a pointer in the correct direction:

#### Explanation:

Once you know that $\frac{d}{\mathrm{dx}} \left(\cot \left(x\right)\right) = - 1 - {\cot}^{2} \left(x\right)$

and you combine it with $\frac{d}{\mathrm{dx}} \left(u v\right) = v \frac{\mathrm{du}}{\mathrm{dx}} + u \frac{\mathrm{dv}}{\mathrm{dx}}$

You are off to a flying start. Just substitute and play until you find a solution:

As a check I put it into Maple which gave: $\left(- 1 - {\cot}^{2} \left\{x\right\}\right) \cos \left(x\right) - \cot \left(x\right) \sin \left(x\right)$

Maple outputs are sometimes a little 'Maple'ish' so need to be translated.

remember that $\cot \left(x\right) = \frac{1}{\tan \left(x\right)} = \frac{\cos \left(x\right)}{\sin \left(x\right)}$
so you may be able to use:

$- \cos \left(x\right) - \frac{{\cos}^{3} \left(x\right)}{{\sin}^{2} \left(x\right)} - \frac{\cos \left(x\right) \sin \left(x\right)}{\sin \left(x\right)}$