# How do you find the derivative of e^(-5x^3+x)?

Mar 14, 2018

$- 15 {e}^{- 5 {x}^{3} + x} {x}^{2} + {e}^{- 5 {x}^{3} + x}$

#### Explanation:

According to the chain rule, $\left(f \left(g \left(x\right)\right)\right) ' = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

Here, we have:

$f \left(u\right) = {e}^{u}$, where

$u = g \left(x\right) = - 5 {x}^{3} + x$

We do:

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

${e}^{u} \cdot \left(- \left(15 {x}^{2} - 1\right)\right)$

${e}^{u} \left(- 15 {x}^{2} + 1\right)$

$- 15 {e}^{u} {x}^{2} + {e}^{u}$

But since $u = - 5 {x}^{3} + x$, we say:

$- 15 {e}^{- 5 {x}^{3} + x} {x}^{2} + {e}^{- 5 {x}^{3} + x}$