Our problem for today is to differentiate ax with the (usual) restrictions that a is a positive number and not equal to 1. The reasoning here is very different from that for finding other derivatives and therefore I hope you and your students find it interesting.
The definition of derivative followed by a little algebra gives tells us that
Since the limit in the expression above is a number, we observe that the derivative of ax is proportional to ax. And also, each value of a gives a different constant. For example if a = 5 then the limit is approximately 1.609438, and so .
I determined this by producing a table of values for the expression in the limit near x = 0. You can do the same using a good calculator, computer, or a spreadsheet.
That’s kind of messy and would require us to find this limit for whatever value of a we were using. It turns out that by finding the value of a for which the limit is 1 we can fix this problem. Your students can do this for themselves by changing the value of a in their table until they get the number that gives a limit of 1.
Okay, that’s going to take a while, but may be challenging. The answer turns out to be close to 2.718281828459045…. Below is the table for this number.
Okay, I cheated. The number is, of course, e. Thus,
The function ex is its own derivative!
And from this we can find the derivatives of all the other exponential functions. First, we define a new function (well maybe not so new) which is the inverse of the function ex called ln(x), the natural logarithm of x. (For more on this see Logarithms.) Then a = eln(a) and ax = (eln(a))x = e(ln(a)x) . Then using the Chain Rule the derivative is
Finally, going back to the first table above where a = 5, we find that the limit we found there 1.609438 = ln(5).
For a video on this topic click here.
Revised 8-28-2018, 6-2-2019