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

Nov 11, 2017

$g ' \left(x\right) = x {\left({x}^{2} - 2\right)}^{\frac{1}{2}} \left(5 {x}^{2} - 7\right)$

#### Explanation:

$\text{given "g(x)=f(x)h(x)" then}$

$g ' \left(x\right) = f \left(x\right) h ' \left(x\right) + h \left(x\right) f ' \left(x\right) \leftarrow \textcolor{b l u e}{\text{product rule}}$

$f \left(x\right) = {x}^{2} - 1 \Rightarrow f ' \left(x\right) = 2 x$

$h \left(x\right) = {\left({x}^{2} - 2\right)}^{\frac{3}{2}}$

$\text{differentiate using the "color(blue)"chain rule}$

$\Rightarrow h ' \left(x\right) = \frac{3}{2} {\left({x}^{2} - 2\right)}^{\frac{1}{2}} \times 2 x = 3 x {\left({x}^{2} - 2\right)}^{\frac{1}{2}}$

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

$\textcolor{w h i t e}{\Rightarrow g ' \left(x\right)} = x {\left({x}^{2} - 2\right)}^{\frac{1}{2}} \left[3 \left({x}^{2} - 1\right) + 2 \left({x}^{2} - 2\right)\right]$

$\textcolor{w h i t e}{\Rightarrow g ' \left(x\right)} = x {\left({x}^{2} - 2\right)}^{\frac{1}{2}} \left(5 {x}^{2} - 7\right)$