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.