# How do you differentiate f(x) = 4(x^2 + x - 1)^10 using the chain rule?

Nov 9, 2015

$f ' \left(x\right) = 40 {\left({x}^{2} + x - 1\right)}^{9} \cdot \left(2 x + 1\right)$

#### Explanation:

let $u = {x}^{2} + x - 1$
now $f \left(x\right) = 4 {u}^{10}$

differentiate:
remember, chain rule requires you to multiply by the derivative of the inside function (or $u '$)
$f ' \left(x\right) = 4 \cdot 10 {u}^{9} \cdot u '$

$f ' \left(x\right) = 4 \cdot 10 {\left({x}^{2} + x - 1\right)}^{9} \cdot \left({x}^{2} + x - 1\right) '$

$f ' \left(x\right) = 4 \cdot 10 {\left({x}^{2} + x - 1\right)}^{9} \cdot \left(2 x + 1\right)$

$f ' \left(x\right) = 40 {\left({x}^{2} + x - 1\right)}^{9} \cdot \left(2 x + 1\right)$

you might rewrite this as:
$f ' \left(x\right) = \left(80 x + 40\right) {\left({x}^{2} + x - 1\right)}^{9}$