# What is the derivative of  f(x) = x^5 * (x^2-3)^6?

Mar 7, 2018

$5 {x}^{4} {\left({x}^{2} - 3\right)}^{6} + 12 {x}^{6} {\left({x}^{2} - 3\right)}^{5}$

#### Explanation:

Here:

$\frac{d}{\mathrm{dx}} {x}^{5} \cdot {\left({x}^{2} - 3\right)}^{6}$

we can use product rule:

$\frac{d}{\mathrm{dx}} \textcolor{red}{a} \cdot \textcolor{b l u e}{b} = \left(\textcolor{red}{a}\right) ' \left(\textcolor{b l u e}{b}\right) + \left(\textcolor{red}{a}\right) \left(\textcolor{b l u e}{b}\right) '$

So:

$\frac{d}{\mathrm{dx}} \textcolor{red}{{x}^{5}} \cdot \textcolor{b l u e}{{\left({x}^{2} - 3\right)}^{6}}$

becomes:

$\left({\textcolor{red}{x}}^{5}\right) ' \textcolor{b l u e}{{\left({x}^{2} - 3\right)}^{6}} + \left({\textcolor{red}{x}}^{5}\right) \left(\textcolor{b l u e}{{\left({x}^{2} - 3\right)}^{6}}\right) '$

Simplifying:

$\left(5 {x}^{4}\right) \textcolor{b l u e}{{\left({x}^{2} - 3\right)}^{6}} + \left({\textcolor{red}{x}}^{5}\right) \left(\textcolor{b l u e}{{\left({x}^{2} - 3\right)}^{6}}\right) '$

$\frac{d}{\mathrm{dx}} \textcolor{b l u e}{{\left({x}^{2} - 3\right)}^{6}}$

We can use chain rule here:

$\frac{d}{\mathrm{dx}} f \left(x\right) = \frac{d}{\mathrm{du}} f \left(u\right) \cdot \frac{d}{\mathrm{dx}} \left(x\right)$

$\to \frac{d}{\mathrm{dx}} {\left({x}^{2} - 3\right)}^{6}$

becomes:

$\frac{d}{\mathrm{dx}} {\left(u\right)}^{6} \cdot \frac{d}{\mathrm{dx}} \left({x}^{2} - 3\right)$

$= 6 {u}^{5} \cdot 2 x$

Since $u = \left({x}^{2} - 3\right)$:

$= 6 {\left({x}^{2} - 3\right)}^{5} \cdot 2 x$

$\frac{d}{\mathrm{dx}} \textcolor{b l u e}{{\left({x}^{2} - 3\right)}^{6}} = 12 x {\left({x}^{2} - 3\right)}^{5}$

Simplifying our former equation:

$\left(5 {x}^{4}\right) \textcolor{b l u e}{{\left({x}^{2} - 3\right)}^{6}} + \left({\textcolor{red}{x}}^{5}\right) \left(\textcolor{b l u e}{{\left({x}^{2} - 3\right)}^{6}}\right) '$

becomes:

$\left(5 {x}^{4}\right) {\left({x}^{2} - 3\right)}^{6} + \left({x}^{5}\right) \left(12 x\right) {\left({x}^{2} - 3\right)}^{5}$

Multiplying it out:

$= 5 {x}^{4} {\left({x}^{2} - 3\right)}^{6} + 12 {x}^{6} {\left({x}^{2} - 3\right)}^{5}$

And there we have our answer