# How do you differentiate g(u) =(u^-2 + u^-3)(u^5 - 2u^2)  using the product rule?

Jan 2, 2018

$3 {u}^{2} + 2 u + \frac{2}{u} ^ 2$

#### Explanation:

The product rule states that

$\frac{d}{\mathrm{dx}} \left(a b\right) = a ' b + a b '$

Let $a = {u}^{-} 2 + {u}^{-} 3$, $b = {u}^{5} - 2 {u}^{2}$

$\therefore g ' \left(u\right) = \left({u}^{-} 2 + {u}^{-} 3\right) ' \left({u}^{5} - 2 {u}^{2}\right) + \left({u}^{-} 2 + {u}^{-} 3\right) \left({u}^{5} - 2 {u}^{2}\right) '$

$g ' \left(u\right) = \left(- \frac{2}{u} ^ 3 - \frac{3}{{u}^{4}}\right) \left({u}^{5} - 2 {u}^{2}\right) + \left({u}^{-} 2 + {u}^{-} 3\right) \left(5 {u}^{4} - 4 u\right)$

$g ' \left(u\right) = - 2 {u}^{2} + \frac{4}{u} - 3 u + \frac{6}{u} ^ 2 + 5 {u}^{2} + 5 u - \frac{4}{u} - \frac{4}{u} ^ 2$

$g ' \left(u\right) = 3 {u}^{2} + 2 u + \frac{2}{u} ^ 2$