# How do you differentiate f(x)=(x^2-2x+1)(x^3-1) using the product rule?

Dec 29, 2017

$5 {x}^{4} - 8 {x}^{3} + 3 {x}^{2} - 2 x + 2$

#### Explanation:

Product Rule $\textcolor{red}{= f ' \left(x\right) g \left(x\right) + f \left(x\right) g ' \left(x\right)}$

In our case, $f \left(x\right) = {x}^{2} - 2 x + 1$ and $g \left(x\right) = {x}^{3} - 1$

Substitute:

$= \frac{d}{\mathrm{dx}} \left({x}^{2} - 2 x + 1\right) \left({x}^{3} - 1\right) + \left({x}^{2} - 2 x + 1\right) \frac{d}{\mathrm{dx}} \left({x}^{3} - 1\right)$

Differentiate:

The Power Rule tells us that $\frac{d}{\mathrm{dx}} {x}^{n} = n {x}^{n - 1}$

$= \left(2 x - 2\right) \left({x}^{3} - 1\right) + \left({x}^{2} - 2 x + 1\right) \left(3 {x}^{2}\right)$

Expand:

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

Simplify:

$= 5 {x}^{4} - 8 {x}^{3} + 3 {x}^{2} - 2 x + 2$