Math Natural exponential function Derivative

# Derivative of the natural exponential functions

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### Remember

By applying the constant factor rule to natural exponential functions, one can say:

$f(x)=a\cdot e^x$ with $a\in\mathbb{R}$
$f'(x)=a\cdot e^x$

There is no other function that possesses this property.

## Derivative of the composition of natural exponential functions

To compute the derivative of compositions of exponential functions, one has to master the differentiation rules, in particular the chain rule.

### Example

$f(x)=e^{x^2-2x}$

1. #### Split function into subfunctions

$g(x)=e^x$ and $h(x)=\color{red}{x^2-2x}$
2. #### Compute derivative of subfunctions

Apply the differentiation rules
$g'(x)=\color{blue}{e}^x$ and $h'(x)=\color{green}{2x-2}$
3. #### Insert

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

$f'(x)=\color{blue}{e}^{\color{red}{x^2-2x}}\cdot \color{green}{(2x-2)}$
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### Hint

Since the derivative of $e^x$ is $e^x$, one only has to calculate the inner derivative when deriving the composition of exponential functions with base e (the outer function remains the same):

$f(x)=e^{g(x)}$

$f'(x)=e^{g(x)}\cdot g'(x)$