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

Dec 27, 2015

$f ' \left(x\right) = - 5 {x}^{4} + 24 {x}^{2} - 6 x - 15$

#### Explanation:

Derivative of product rule

Given $\text{ " } h = f \cdot g$

$h ' = f g ' + f ' g$

The original problem

$f \left(x\right) = \left(5 - {x}^{2}\right) \left({x}^{3} - 3 x + 3\right)$

$f ' \left(x\right) = \left(5 - {x}^{2}\right) \frac{d}{\mathrm{dx}} \left({x}^{3} - 3 x + 3\right) + \frac{d}{\mathrm{dx}} \left(5 - {x}^{2}\right) \left({x}^{3} - 3 x + 3\right)$

$\implies \left(5 - {x}^{2}\right) \left(3 {x}^{2} - 3\right) + \left(- 2 x\right) \left({x}^{3} - 3 x + 3\right)$

Now we can multiply and combine like terms

$\implies \left(15 {x}^{2} - 15 - 3 {x}^{4} + 3 {x}^{2}\right) + \left(- 2 {x}^{4} + 6 {x}^{2} - 6 x\right)$
$\implies - 5 {x}^{4} + 24 {x}^{2} - 6 x - 15$