Derivatives of exponential functions

When computing the derivative of general exponential functions one uses the natural logarithm.

Example

$f(x)=2^x$
$f'(x)=2^x\cdot\ln(2)$

Derivation

Here the derivation of the phrase is described.

We are looking for the derivative of $f(x)=a^x$

1. Rewrite as exponential function

   Since the ln-functionis the inverse of the exponential function, the following applies:

   $x=e^{\ln(x)}$
   $a^x=e^{\ln(a^x)}$

   Now the logarithm law for powers is applied.
   $a^x=e^{x\cdot\ln(a)}$

   $f(x)=a^x=e^{x\cdot\ln(a)}$

2. Compute the derivative of the exponential function

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

   $\ln(a)$is a constant factor (constant factor rule) and $(x)'=1$

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

3. Rewrite exponential function

   Apply the method from the first step backwards:

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