 In this video, I want to talk about the derivative of the natural exponential function edx. Have you ever wondered why mathematicians like the number e, which is approximately, it's an irrational number, right? It's approximately 2.718281828. What's the big deal? What's so natural about this exponential? I mean, after all, we call the logarithm base e, which we usually call ln for short, the natural log or the logarithm natural. What's so natural about this number e? Turns out this video will actually answer that question. From an algebraic point of view, there's nothing really natural about e. But from a calculus point of view, everything is natural about e. It's the best exponential. The function e to the x has the property that it's equal to its own derivative. The derivative of e to the x with respect to x is e to the x itself. If you take the derivative of e to the x, it's just e to the x. It's its own derivative. With respect to the differential operator, e to the x is a fixed point. It doesn't move when you take its derivative, which is a very interesting concept. If you think about it graphically, that is to say, we were to graph the function e to the x, it would look something like the following. What we're saying here is if you take a tangent line at any point on the graph, the slope of the tangent line is actually identical to its height above the x-axis. It's very phenomenal. We'll see later on why that's such an important thing, but let's actually see the proof first. If we want to take the derivative e to the x, we have to look at a difference quotient. Use the definition of the derivative. It's going to be the limit as h approaches 0 of e to the x plus h. Notice if f of x here is equal to e to the x, then this right here is just f of x plus h. That's what it is. So we're taking the limit of e to the x plus h minus e to the x all over h right here. So we have our difference quotient, but it's an exponential expression on top, right? You have e to the x plus h minus e to the x. By exponent rules, you can take e to the x plus h and factor it as e to the x times e to the h. The benefit of this is that now you see that e to the x is a common factor, which we can factor out giving this right here. And when you look at the expression e to the x, it doesn't depend on h whatsoever. e to the x is a constant with respect to this limit process. As h is getting close to 0, e to the x is just still e to the x. So as such, we can factor it out as constant multiple. So this difference quotient turns into e to the x times the limit as h approaches 0 of e to the h minus 1 over h. And now this is the hard part. This limit of e to the h minus 1 over h as h approaches 0, this would normally be a difficult thing to do. But the good news is in our lecture series, we've already approached this limit before. In a previous video, we showed that this limit is equal to 1 as a consequence of the squeeze theorem. Take a look at the link that you saw on the screen if you want to see the details of that. What that does for us then is that if we finish this up, e to the x times this limit, which turns out to be 1, you'll get e to the x times 1, and we see that then the derivative of e to the x is equal to e to the x itself. It's quite phenomenal, really. So e to the x is its own derivative. The derivative of e to the x is e to the x. This is incredible, really is. You're gonna love this because it makes calculating the derivative involving e to the x so much easier. But math geeks like me also really love this because this property will be very valuable in the process of solving so-called differential equations, which is something we do in calculus all the time. So in a nutshell, this is an awesome property. In fact, the reason mathematicians first started considering the number e is because they wanted a function which exhibited the property that when you take its derivative, you get back the original function. And that's where essentially the number e came from. Let's put this into practice. Let's find the derivative of f of x equals e to the x minus x. Let's actually find the first derivative, the second derivative. We can keep on going, right? Because the derivative of e to the x is always itself, it makes it very easy to calculate these higher derivatives. If we take the derivative of f of x, we're going to take the derivative of e to the x minus x. But by the sum property, we'll take the derivative of e to the x separate from x, for which the derivative of x is going to be one and the derivative of e to the x is itself. So the derivative f prime is going to be e to the x minus one. If we then take the second derivative, we're going to take the derivative of the first derivative, which then gives us the derivative of e to the x is e to the x. The derivative of one is going to be zero, and so the derivative becomes e to the x. And then we see after this that all higher derivatives, like the third derivative, it'll be e to the x and all higher derivatives are likewise going to be e to the x, in which case it's a very simple calculation of the derivative of e to the x, it's always itself.