What is the derivative of (x-1)(x^2+2)^3?

May 20, 2018

${\left({x}^{2} + 2\right)}^{2} \left(7 {x}^{2} - 6 x + 2\right)$

Explanation:

$\text{differentiate using the "color(blue)"product/chain rules}$

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

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

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

$\frac{\mathrm{dy}}{\mathrm{dx}} = f ' \left(g \left(x\right)\right) \times g ' \left(x\right) \leftarrow \textcolor{b l u e}{\text{chain rule}}$

$g \left(x\right) = x - 1 \Rightarrow g ' \left(x\right) = 1$

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

$\textcolor{w h i t e}{\times \times \times \times \times \times \times \times \times} = 6 x {\left({x}^{2} + 2\right)}^{2}$

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

$= 6 x \left(x - 1\right) {\left({x}^{2} + 2\right)}^{2} + {\left({x}^{2} + 2\right)}^{3}$

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

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