# How do you find the derivative of cos(1-2x)^2?

Feb 9, 2016

$\frac{d}{\mathrm{dx}} {\cos}^{2} \left(1 - 2 x\right) = 4 \sin \left(1 - 2 x\right)$

#### Explanation:

Use the chain rule :

$\frac{d}{\mathrm{dx}} \left(f \circ g\right) \left(x\right) = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

A first iteration of the chain rule reduces the exponent of the cosine using the power rule.

$\frac{d}{\mathrm{dx}} {\cos}^{2} \left(1 - 2 x\right) = 2 \frac{d}{\mathrm{dx}} \cos \left(1 - 2 x\right)$

A second iteration of the chain rule changes the trig function using the identity, $\frac{d}{\mathrm{dx}} \cos x = - \sin x$.

$2 \frac{d}{\mathrm{dx}} \cos \left(1 - 2 x\right) = 2 \left(- \sin \left(1 - 2 x\right)\right) \frac{d}{\mathrm{dx}} \left(1 - 2 x\right)$

Lastly, apply the power rule to the remaining derivative statement.

$- 2 \sin \left(1 - 2 x\right) \frac{d}{\mathrm{dx}} \left(1 - 2 x\right) = - 2 \sin \left(1 - 2 x\right) \left(- 2\right)$

$= 4 \sin \left(1 - 2 x\right)$