# How do you differentiate f(x) = (−7 x^2 − 5)^8 (2 x^2 − 9)^9 ?

$\textcolor{red}{f ' \left(x\right) = 4 x {\left(7 {x}^{2} + 5\right)}^{7} \cdot {\left(2 {x}^{2} - 9\right)}^{8} \left(119 {x}^{2} - 207\right)}$

#### Explanation:

For the given function $f \left(x\right) = {\left(- 7 {x}^{2} - 5\right)}^{8} {\left(2 {x}^{2} - 9\right)}^{9}$,
we are going to use the formula $\frac{d}{\mathrm{dx}} \left(u v\right) = u \frac{d}{\mathrm{dx}} \left(v\right) + v \cdot \frac{d}{\mathrm{dx}} \left(u\right)$

Let $u = {\left(- 7 {x}^{2} - 5\right)}^{8} = {\left(- 1\right)}^{8} {\left(7 {x}^{2} + 5\right)}^{8} = {\left(7 {x}^{2} + 5\right)}^{8}$ and $v = {\left(2 {x}^{2} - 9\right)}^{9}$

$f ' \left(x\right) = {\left(7 {x}^{2} + 5\right)}^{8} \frac{d}{\mathrm{dx}} {\left(2 {x}^{2} - 9\right)}^{9} + {\left(2 {x}^{2} - 9\right)}^{9} \cdot \frac{d}{\mathrm{dx}} {\left(7 {x}^{2} + 5\right)}^{8}$

$f ' \left(x\right) = {\left(7 {x}^{2} + 5\right)}^{8} \cdot 9 {\left(2 {x}^{2} - 9\right)}^{9 - 1} \frac{d}{\mathrm{dx}} \left(2 {x}^{2} - 9\right) + {\left(2 {x}^{2} - 9\right)}^{9} \cdot 8 {\left(7 {x}^{2} + 5\right)}^{8 - 1} \frac{d}{\mathrm{dx}} \left(7 {x}^{2} + 5\right)$

$f ' \left(x\right) = {\left(7 {x}^{2} + 5\right)}^{8} \cdot 9 {\left(2 {x}^{2} - 9\right)}^{8} \cdot \left(4 x - 0\right) + {\left(2 {x}^{2} - 9\right)}^{9} \cdot 8 {\left(7 {x}^{2} + 5\right)}^{7} \left(14 x + 0\right)$

$f ' \left(x\right) = {\left(7 {x}^{2} + 5\right)}^{8} \cdot 9 {\left(2 {x}^{2} - 9\right)}^{8} \cdot \left(4 x\right) + {\left(2 {x}^{2} - 9\right)}^{9} \cdot 8 {\left(7 {x}^{2} + 5\right)}^{7} \left(14 x\right)$

Factoring common factors

$f ' \left(x\right) =$
$4 x {\left(7 {x}^{2} + 5\right)}^{7} \cdot {\left(2 {x}^{2} - 9\right)}^{8} \left[\left(7 {x}^{2} + 5\right) \cdot 9 + \left(2 {x}^{2} - 9\right) \cdot 2 \left(14\right)\right]$

$f ' \left(x\right) = 4 x {\left(7 {x}^{2} + 5\right)}^{7} \cdot {\left(2 {x}^{2} - 9\right)}^{8} \left[\left(63 {x}^{2} + 45\right) + \left(56 {x}^{2} - 252\right)\right]$

$\textcolor{red}{f ' \left(x\right) = 4 x {\left(7 {x}^{2} + 5\right)}^{7} \cdot {\left(2 {x}^{2} - 9\right)}^{8} \left(119 {x}^{2} - 207\right)}$

God bless....I hope the explanation is useful.