# What is the derivative of f(x) = x^5 * (x^2-3)^?

Apr 21, 2017

$7 {x}^{6} - 15 {x}^{4}$

#### Explanation:

Using the product rule:

$\frac{d}{\mathrm{dx}} \left(u w\right) = u ' w + u w '$

This basically means in order to find the derivative of the function $\left(u w\right)$, you have to separate it into 2 separate functions, $u$ and $w$. Then you find the derivative of $u$ and $w$ and multiply their derivatives $u '$ and $w '$ with the other function and add their products together.

Where
$\textcolor{w h i t e}{a a a a a a a}$(u)$\textcolor{w h i t e}{a a a a a a}$(w)
$\textcolor{w h i t e}{a a a a a a a}$$\downarrow$$\textcolor{w h i t e}{a a a a a}$$\downarrow$
$f \left(x\right) = \left({x}^{5}\right) \cdot \left({x}^{2} - 3\right)$

$u = {x}^{5}$$\textcolor{w h i t e}{a a a a a a}$$w = \left({x}^{2} - 3\right)$
$u ' = 5 {x}^{4}$$\textcolor{w h i t e}{a a a a}$$w ' = 2 x$

$- - - - - - - - - - - - - - - - - - -$

$= \left(5 {x}^{4}\right) \left({x}^{2} - 3\right) + \left(2 x\right) \left({x}^{5}\right)$
$= 5 {x}^{6} - 15 {x}^{4} + 2 {x}^{6}$
$= 7 {x}^{6} - 15 {x}^{4}$