# How do you differentiate g(x) =(x^2 + 6) (x^2 - 2x + 1)^4  using the product rule?

Jan 9, 2017

$g \left(x\right) = \left({x}^{2} + 6\right) {\left({x}^{2} - 2 x + 1\right)}^{4}$

Our task will be simpler if we note that ${x}^{2} - 2 x + 1 = {\left(x - 1\right)}^{2}$, so

$g \left(x\right) = \left({x}^{2} + 6\right) {\left(x - 1\right)}^{8}$

$g ' \left(x\right) = 2 x {\left(x - 1\right)}^{8} + \left({x}^{2} + 6\right) \cdot 8 {\left(x - 1\right)}^{7}$

$= 2 {\left(x - 1\right)}^{7} \left[x \left(x - 1\right) + 4 \left({x}^{2} + 6\right)\right]$

$= 2 {\left(x - 1\right)}^{7} \left(5 {x}^{2} - x + 24\right)$