# If f(x)= cos5 x  and g(x) = e^(3+4x ) , how do you differentiate f(g(x))  using the chain rule?

Jan 5, 2016

Leibniz's notation can come in handy.

#### Explanation:

$f \left(x\right) = \cos \left(5 x\right)$

Let $g \left(x\right) = u$. Then the derivative:

$\left(f \left(g \left(x\right)\right)\right) ' = \left(f \left(u\right)\right) ' = \frac{\mathrm{df} \left(u\right)}{\mathrm{dx}} = \frac{\mathrm{df} \left(u\right)}{\mathrm{dx}} \frac{\mathrm{du}}{\mathrm{du}} = \frac{\mathrm{df} \left(u\right)}{\mathrm{du}} \frac{\mathrm{du}}{\mathrm{dx}} =$

$= \frac{\mathrm{dc} o s \left(5 u\right)}{\mathrm{du}} \cdot \frac{d \left({e}^{3 + 4 x}\right)}{\mathrm{dx}} =$

$= - \sin \left(5 u\right) \cdot \frac{d \left(5 u\right)}{\mathrm{du}} \cdot {e}^{3 + 4 x} \frac{d \left(3 + 4 x\right)}{\mathrm{dx}} =$

$= - \sin \left(5 u\right) \cdot 5 \cdot {e}^{3 + 4 x} \cdot 4 =$

$= - 20 \sin \left(5 u\right) \cdot {e}^{3 + 4 x}$