# How do you find the derivative of 3(x^2-2)^4?

Jun 4, 2015

We can use the chain rule, which states that $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \frac{\mathrm{du}}{\mathrm{dx}}$.

To do so, we'll rename $u = {x}^{2} - 2$, thus rewriting the expression as $y = 3 {u}^{4}$

Following chain rule's statements:

$\frac{\mathrm{dy}}{\mathrm{du}} = 12 {u}^{3}$
$\frac{\mathrm{du}}{\mathrm{dx}} = 2 x$

Combining them:

$\frac{\mathrm{dy}}{\mathrm{dx}} = 12 {u}^{3} \left(2 x\right)$. Now, we need to substitute $u$.

$\frac{\mathrm{dy}}{\mathrm{dx}} = 12 {\left({x}^{2} - 2\right)}^{3} \left(2 x\right) = \textcolor{g r e e n}{24 x {\left({x}^{2} - 2\right)}^{3}}$