Math Derivative rules Chain rule

# Chain rule

The chain rule must be used when deriving chained functions. A chained function is a function of a function.

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### Remember

$f(x)=g(h(x))$

$f'(x)=g'(h(x))\cdot h'(x)$

$g(x)$ is the outer function.
$g'(x)$ is the outer derivative.
$h(x)$ is the inner function.
$h'(x)$ is the inner derivative.

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### Hint

The chain rule is often used to derive elementary functions.

### Examples

$f(x)=(\color{red}{x^3+4})^5$

1. #### Split function into subfunctions

$g(x)=x^5$ and $h(x)=\color{red}{x^3+4}$
2. #### Derive subfunctions

Apply the power rule
$g'(x)=\color{blue}{5}x\color{blue}{^4}$ and $h'(x)=\color{green}{3x^2}$
3. #### Insert

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

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

$f(x)=\sin(\color{red}{x^5})$

1. #### Split function into subfunctions

$g(x)=\sin(x)$ and $h(x)=\color{red}{x^5}$
2. #### Derive subfunctions

Derive sine function and apply the power rule
$g'(x)=\color{blue}{\cos}(x)$ and $h'(x)=\color{green}{5x^4}$
3. #### Insert

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

$f'(x)=\color{blue}{\cos}(\color{red}{x^5})\cdot \color{green}{5x^4}$