Section Summary

In this section we have developed formulas for differentiating exponential functions. In particular, we found that

d d ⁢ t ⁢ b t = ( ln ⁢ b ) b t.

The logarithm in this formula, $\ln$, is the natural logarithm, the one that has Euler's number $e$ as its base. (An approximate value of $e$ is $2.718281828$.) When $e$ is also the exponential base, we get the simpler formula

d d ⁢ t ⁢ e t = e t.

This natural exponential function, the one that is its own derivative, is also called $\exp$: that is, $\exp t = e^t$.

More generally, we found a whole family of functions that all have derivatives proportional to themselves:

d d t A ⁢ e k ⁢ t = k ⁢ A ⁢ e k ⁢ t.