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, `text(ln) text(,)` 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 `text(exp)`: that is, `text(exp)text[(] t text[)] = 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 .