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):

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

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