Chain rule

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

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

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

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