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Proof of the Derivative of $a^x$

The derivative of $f\left(x\right)=a^x$ is $\displaystyle f'(x)=a^x \left(\ln a\right)$.
To prove this, we will use the following properties:
  1. $x$ can be written as $e$
  2. $m \ln x= \ln$
  3. Using the chain rule, the derivative of $e^{f(x)}$ is $e^{f(x)} \Bigl($ $\Bigl)$
Use property 1:
$f(x)=a^x=e$

Use property 2:
$f(x)=e$ $\ln a$

Find $f'(x)$ using the chain rule (3):
$f'(x)=e^{x \ln a} \Bigl($ $\Bigl)$

Write $e^{x \ln a}$ as $a^x$:
$f'(x)=$