Math Derivative rules Product rule

# Product rule

In order to derive a product, you use the product rule.

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### Remember

$f(x)=g(x)\cdot h(x)$
$f'(x)=\color{red}{g'(x)}\cdot h(x) +\color{red}{h'(x)}\cdot g(x)$

### Example

$f(x)=x^2\cdot3x^4$

1. #### Split function into subfunctions

$g(x)=x^2$ and $h(x)=3x^4$
2. #### Derive subfunctions

$g'(x)=\color{blue}{2x}$ and $h'(x)=\color{green}{12x^3}$
3. #### Insert

$f'(x)=\color{blue}{g'(x)}\cdot h(x) +\color{green}{h'(x)}\cdot g(x)$

$f'(x)=\color{blue}{2x}\cdot 3x^4 +\color{green}{12x^3}\cdot x^2$ $=6x^5+12x^5$ $=18x^5$

A tip: This example can also be calculated and derived only with the power rule:

$f(x)=x^2\cdot3x^4=3x^6$
$f'(x)=18x^5$