Check out this dialogue between two calculus students (based on a true story):
- Devyn
- We learned how to take the derivative of , but I wonder how you find the derivative of ?
- Riley
- That’s easy,
- Devyn
- I don’t think it is that easy, but you could be right.
- Riley
- Could be?
To get an idea of how to find the derivative of an exponential function, let’s look at one specific exponential function:
Remember from the definition of the derivative that the derivative of a function at a value is approximated by slopes of secant lines. Let’s look at what is happening with .
Recall that the slope of the secant line between and is
Looking at these, it seems that the derivative of is almost . It turns out that it is a constant multiple of .
If we try the same process with an exponential function to a different base, we will see the same thing occur. Again, the derivative ends up being a constant multiple of the original function.
What is this constant multiple? We will find out in the next few pages, but first, is there a basic exponential function where the constant is exactly 1? There is and we will look at this function next.