Math Natural exponential function Derivatives of exponential functions

Derivatives of exponential functions

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







When computing the derivative of an exponential function, it shifts along the x-axis.


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:


    Now the logarithm law for powers is applied.

  2. Compute the derivative of the exponential function

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


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

  3. Rewrite exponential function

    Apply the method from the first step backwards: