# How do you find the derivative of y = (x - 2)^3 (x^2 + 9)^4?

Aug 6, 2017

$y ' \left(x\right) = 3 {\left({x}^{2} + 9\right)}^{4} {\left(x - 2\right)}^{2} + 8 x {\left(x - 2\right)}^{3} {\left({x}^{2} + 9\right)}^{3}$

#### Explanation:

We're asked to find the derivative

$\frac{\mathrm{dy}}{\mathrm{dx}} \left[y = {\left(x - 2\right)}^{3} {\left({x}^{2} + 9\right)}^{4}\right]$

Let's first use the product rule, which is

$\frac{d}{\mathrm{dx}} \left[u v\right] = v \frac{\mathrm{du}}{\mathrm{dx}} + u \frac{\mathrm{dv}}{\mathrm{dx}}$

where

• $u = {\left(x - 2\right)}^{3}$

• $v = {\left({x}^{2} + 9\right)}^{4}$:

$y ' \left(x\right) = {\left({x}^{2} + 9\right)}^{4} \frac{d}{\mathrm{dx}} \left[{\left(x - 2\right)}^{3}\right] + {\left(x - 2\right)}^{3} \frac{d}{\mathrm{dx}} \left[{\left({x}^{2} + 9\right)}^{4}\right]$

Now, we use the chain rule for the first term:

$\frac{d}{\mathrm{dx}} \left[{\left(x - 2\right)}^{3}\right] = \frac{d}{\mathrm{du}} \left[{u}^{3}\right] \frac{\mathrm{du}}{\mathrm{dx}}$

where

• $u = x - 2$

• $\frac{d}{\mathrm{du}} \left[{u}^{3}\right] = 3 {u}^{2}$:

$y ' \left(x\right) = {\left({x}^{2} + 9\right)}^{4} 3 {\left(x - 2\right)}^{2} \frac{d}{\mathrm{dx}} \left[x - 2\right] + {\left(x - 2\right)}^{3} \frac{d}{\mathrm{dx}} \left[{\left({x}^{2} + 9\right)}^{4}\right]$

The derivative of $x - 2$ is $1$:

$y ' \left(x\right) = 3 {\left({x}^{2} + 9\right)}^{4} {\left(x - 2\right)}^{2} + {\left(x - 2\right)}^{3} \frac{d}{\mathrm{dx}} \left[{\left({x}^{2} + 9\right)}^{4}\right]$

Now we use the chain rule on the second term:

$\frac{d}{\mathrm{dx}} \left[{\left({x}^{2} + 9\right)}^{4}\right] = \frac{d}{\mathrm{du}} \left[{u}^{4}\right] \frac{\mathrm{du}}{\mathrm{dx}}$

where

• $u = {x}^{2} + 9$

• $\frac{d}{\mathrm{du}} \left[{u}^{4}\right] = 4 {u}^{3}$:

$y ' \left(x\right) = 3 {\left({x}^{2} + 9\right)}^{4} {\left(x - 2\right)}^{2} + {\left(x - 2\right)}^{2} 4 {\left({x}^{2} + 9\right)}^{3} \frac{d}{\mathrm{dx}} \left[{x}^{2} + 9\right]$

The derivative of ${x}^{2} + 9$ is $2 x$:

color(blue)(ulbar(|stackrel(" ")(" "y'(x) = 3(x^2+9)^4(x-2)^2 + 8x(x-2)^2 (x^2+9)^3" ")|)