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

The derivative is

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

Feb 5, 2016

Remember that ${\cos}^{2} x = {\left(\cos x\right)}^{2}$.

#### Explanation:

Now use $\frac{d}{\mathrm{dx}} \left({u}^{2}\right) = 2 u \frac{\mathrm{du}}{\mathrm{dx}}$.

In this case $u = \cos x$, so that $\frac{\mathrm{du}}{\mathrm{dx}} = - \sin x$

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

Depending on how well you know trigonometry, you may or may not recognize $2 \sin x \cos x$ as equal to $\sin 2 x$.

We can also write:

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