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

Mar 7, 2017

$y ' \left(u\right) = 3 {u}^{2} + 2 u - 4 {u}^{-} 1 - 4 {u}^{-} 2 + 4 {u}^{-} 5 + 6 {u}^{-} 6$

#### Explanation:

differentiate using the $\textcolor{b l u e}{\text{product rule}}$

$\text{Given "y(u)=g(u).h(u)" then}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y ' \left(u\right) = g \left(u\right) h ' \left(u\right) + h \left(u\right) g ' \left(u\right)} \textcolor{w h i t e}{\frac{2}{2}} |}}} \leftarrow \text{ product rule}$

$\text{here } g \left(u\right) = {u}^{-} 2 + {u}^{-} 3 \Rightarrow g ' \left(u\right) = - 2 {u}^{-} 3 - 3 {u}^{-} 4$

$\text{and } h \left(u\right) = {u}^{5} - 2 {u}^{-} 2 \Rightarrow h ' \left(u\right) = 5 {u}^{4} - 4 u$

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

We can 'tidy up' by distributing the products.

$= 5 {u}^{2} - 4 {u}^{-} 1 + 5 u - 4 {u}^{-} 2 - 2 {u}^{2} - 3 u + 4 {u}^{-} 5 + 6 {u}^{-} 6$

$= 3 {u}^{2} + 2 u - 4 {u}^{-} 1 - 4 {u}^{-} 2 + 4 {u}^{-} 5 + 6 {u}^{-} 6$