The derivative of an exponential function can be derived using the definition of the derivative. Derivatives of exponential functions involve the natural logarithm function, which itself is an important limit in Calculus, as well as the initial exponential function. The derivative is the natural logarithm of the base times the original function.
We've been talking about derivatives of different kinds of functions well one class of functions that's really important is exponential functions. Now recall that exponential function is one of the form f of x equals a to the x where a is a positive constant but not 1. Let's find this derivative f prime of x now you'll recall that usually the first thing we do is we work on the difference quotient f of x plus h minus f of x over h. Now for this function f of x plus h is a to the x plus h and f of x is a to the x. So we just have to simplify this just a little bit. Now a to the x plus h is the same as a to the x times a to the h by a property of exponents, minus a to the x, and now you can see that a to the x is a common factor in both terms of the numerator and it could be factored out. So I can write this as and I'll pull it out to the right and what's left? In this term a to the h minus and then 1 all over h, so this is my simplified difference quotient.
Let me pop that up into the limit because we'll call it the derivative of f is the limit as h approached zero of the difference quotient. So this will be the limit as h approaches zero of a to the h minus 1 over h times a to the x. Now one of the things that we showed in a previous episode was that the limit of this quantity was actually the natural log of a. Now this actually is going to be constant with respect to h, as h goes to zero nothing happens to a to the x but this thing approaches ln of a and so thats our derivative. f prime of x equals lna and so we summarize this by saying the derivative of an exponential function, the derivative with respect to x of a to the x equals natural log of a times a to the x. So it's kind of interesting every exponential functions derivative is just a constant times that function.