# How do you differentiate f(x)=(cos^2x^2)^(7/3) using the chain rule?

Sep 12, 2016

$- \frac{28}{3} x {\cos}^{\frac{11}{3}} \left({x}^{2}\right) \sin \left({x}^{2}\right)$

#### Explanation:

$f \left(x\right) = {\left({\cos}^{2} {x}^{2}\right)}^{\frac{7}{3}}$

$= \left({\cos}^{\frac{14}{3}} {x}^{2}\right)$

$f ' \left(x\right) = \frac{14}{3} \cdot {\cos}^{\frac{11}{3}} {x}^{2} \cdot \frac{d}{\mathrm{dx}} \left(\cos {x}^{2}\right)$ (Power rule and Chain rule)

$= \frac{14}{3} \cdot {\cos}^{\frac{11}{3}} {x}^{2} \cdot \left(- \sin {x}^{2}\right) \cdot 2 x$ (Chain rule)

$= - \frac{28}{3} x {\cos}^{\frac{11}{3}} \left({x}^{2}\right) \sin \left({x}^{2}\right)$