# How do you find the derivative of cos (2x)?

Jan 27, 2016

$f ' \left(x\right) = - 2 \sin \left(2 x\right)$

#### Explanation:

You need to apply the chain rule:

$f \left(x\right) = \cos \left(\textcolor{b l u e}{2 x}\right) = \cos \left(\textcolor{b l u e}{u}\right) \text{ where } u = 2 x$

Thus, you need to differentiate $\cos u$ and you need to differentiate $2 x$ and multiply those derivatives to obtain the derivative of $f \left(x\right)$:

$f ' \left(x\right) = \left[\cos u\right] ' \cdot \left[u\right] ' = \left[\cos u\right] ' \cdot \left[2 x\right] '$

$= - \sin \textcolor{b l u e}{u} \cdot 2 = - \sin \left(\textcolor{b l u e}{2 x}\right) \cdot 2 = - 2 \sin \left(2 x\right)$