 Let's find the derivatives of the following functions that involve exponential expressions of some kind. So for example, consider the function y equals e to the five x power. Now, one thing you're gonna commonly see when it comes to derivatives involving exponentials is you're commonly gonna have to use the chain rule, which deals with taking derivatives of functions that are composed together. And for exponential functions, it's very common to consider that the x one itself is this inner function. We put five x inside of the outer function, which is e to the x. So we can see this here, e to the u composed with the inner function five to the x here. This is the function in play here, where u is just our inner function, five to the x. So if we put five the x in for u, we have an exponential function on the outside. We have an inner function. In this case, it's five x there. When we take the derivative using the chain rule, the chain rule is trying to tell us that the derivative of y with respect to x is gonna equal the so-called outer derivative, which is dy over du, times the so-called inner derivative, which is du over dx. So we have two factors. There's the inner derivative, which we need to have, and we also have the outer derivative. You don't wanna forget the inner derivative, very common mistake there. You don't wanna do it. So calculating the derivative here, dy over dx, we have to take the outer derivative first. So we take the derivative of e to the u. So we're gonna get something like e to the u prime, and then we times that by the inner derivative, which will look like five to the x prime, for which when you take the derivative of the natural exponential, you go back itself. This is something we've seen before. The derivative of e to the u is e to the u. And when you take the derivative of five x with respect to x, you're gonna get back a five. So this looks like we should be getting five e to the u. But remember, u itself was this artificial variable that we introduced to help us see the decomposition here. U, of course, is just equal to the inner function five, five x there. So the final derivative would look like five times e to the five x, like so. Let's take a look at another example. How can we calculate the derivative using the chain rule of the function y equals 10 e to the three x squared? Well, if we're trying to calculate the derivative y prime, we have to take the derivative of 10 e to the three x squared, of course, by derivative properties, we can actually factor out the constant multiple out of the process. So we have to take e to the three x squared power and find its derivative. The decomposition is gonna be similar till we've solved a moment ago. The exponent itself is the inner function. The outer function is going to be the exponential, the natural exponential here. For which case, when you take the derivative of the natural exponential, you get back itself. The derivative of e to the x is e to the x. And so in this case, when you take the derivative of e to the three x squared, you're gonna get back e to the three x squared. That part is gonna be consistent because e to the x is its own derivative. But what we must not forget is we need to have the inner derivative as well. That is, we need to have the derivative of three x squared in order for this to be correct. For which, of course, we know by usual derivative calculations, the derivative of three x squared is going to be a six x by the power rule. Six times 10 is equal to 60. I'm also gonna throw the x in front. So we get 60x times e to the three x squared, which would be the derivative of this exponential using the chain rule. Let's look at one more example of taking a derivative of the exponential function. This time, let's compose an exponential with a trigonometric function. The exponent itself, sine of x, this is our inner function. And then the outer functions can be the e to the x again, right? And so by the chain rule, y prime is gonna look like the outer derivative. What's the derivative of the natural exponential? It's itself. So we get e to the sine of x power. But then don't forget the inner derivative. We're gonna get the derivative of sine, which we know to be cosine. So our derivative here would look like e to the sine of x times cosine of x. For which there's really not much more you can do to simplify that. We end up with our derivative right here. And so this gives us some examples of computing derivatives of exponential functions using the chain rule, which is something we often have to use when we have to calculate derivatives of exponential functions.