# How do you differentiate f(x)=cos(e^(x^4))  using the chain rule?

Jun 18, 2016

$\frac{\mathrm{df}}{\mathrm{dx}} = - 4 {x}^{3} {e}^{{x}^{4}} \sin \left({e}^{{x}^{4}}\right)$

#### Explanation:

Here $f \left(x\right) = \cos \left(g \left(x\right)\right)$, where $g \left(x\right) = {e}^{h \left(x\right)}$ and $h \left(x\right) = {x}^{4}$

Hence, using chain rule

$\frac{\mathrm{df}}{\mathrm{dx}} = - \sin \left({e}^{{x}^{4}}\right) \times {e}^{{x}^{4}} \times 4 {x}^{3}$

or $\frac{\mathrm{df}}{\mathrm{dx}} = - 4 {x}^{3} {e}^{{x}^{4}} \sin \left({e}^{{x}^{4}}\right)$