 Hi, I'm Zor. Welcome to Unizor Education. I will continue talking about certain fundamental functions and how to take the derivative from these functions. This lecture is part of the advanced course of mathematics for teenagers and high school students. It's presented on Unizor.com, and I do recommend you to watch this lecture from this website because there are very nice notes for each lecture, including this one, and exams for those who are interested in getting a little bit deeper into this. So today we'll talk about exponential functions and how to take the derivative from it. Well, let's consider the basic exponential function e to the power of x. Now, you remember how I introduced the number e. Basically, I was talking graphically about tangent at that particular point, and I said that, well, if it's 2 to the power of x, then this would be less than 45 degrees. If it's 3 to the power of x, it's greater than 45 degrees. But 45 degrees would be exactly at some number which I have called e, and the approximate value is 2.71, and it's actually irrational number, so it's infinite number of decimal signs here. All right, so that's the definition. Now, let's talk about how can we calculate what's the value of the derivative of this function? Okay, here it is. Now, as you remember, the definition of the derivative is limit f at x plus delta x minus f at x divided by delta x will delta x tends to 0. Now, our f at x is this. So we are talking about e to the power x plus delta x minus e to the power of x divided by delta x, right? Now, what is the addition of the exponential part here? Well, that's basically a multiplication because if you remember, a to the power of b plus c is a to the power of b times a to the power of c, right? That's basically one of the fundamental properties of exponential functions. So in this case, I can write that this is equal to e to the power of x times e to the power delta x minus e to the power of x divided by delta x, which is equal to, we can factor out e to the power of x. Now, what happens when e to the power of some infinitesimal value minus 1 divided by this infinitesimal value, whenever this infinitesimal value actually goes to 0, we have this as one of the very important limits, which looks like this. Limit e to the power of, let's say, a minus 1 divided by a as a goes to 0 is equal to 0. Sorry, 1. Remember this? And again, geometrical sense of this is the same as I was talking about. This is 45 degrees, right? So e to the power of a is this. If this is a, so this difference, this is 1, right? So e to the power of a with a is equal to 0 goes through 0.1. If a is equal here, then this is my e to the power of a. And if I will subtract minus 1, it means this part divided by this part. And as soon as a goes to 0, this tangential line goes to 45 degrees, which means that this should be equal to this and the ratio should be equal to 1. That's geometrical sense of it. So this is a known limit which we have discussed in one of the previous lectures. So as delta x goes to 0, this thing goes to 1. And the whole thing tends to e to the power of x. And here is the wonderful thing about this, that for this particular function its derivative is itself. So e to the power of x is the only function, actually, which has this particular property. Its derivative is equal to itself. It's a very interesting property and I was kind of very much surprised when I first learned about that. But that's the only function. That's why it's such an important number, e, in calculus and mathematics in general, because e to the power of x is the only function which actually has this property of being the same as its own derivative. And by the way, this function is defined for all real x. And obviously its derivative also has the same domain, all real numbers. All right, so that's e to the power of x. What other exponential function we can actually take derivative from? Well, another exponential function is the general one, which means f of x is equal to e to the power of x, where a is some positive number. Any positive number, okay? Now, so what should I know right now? Again, a to the power x plus delta x minus a to the power of x divided by delta x. And we have to find the limit of this as delta x goes to 0. Well, we will do the same, this is multiplication of a to the power of x times a to the power of delta x. So it's a to the power of x and that would be the... Now, what is the limit of this as delta x goes to 0? Let's just think about it. A to the power of delta x is the same as e to the power of natural logarithm a of a to the power of delta x, right? A is the same as e to the power of natural logarithm of a. That's basically a definition of natural logarithm, if you remember. Now, since this is true, what happens if I'm raising in one power and then the result raising in another power? It's the multiplication of powers. Again, known property of the powers. So, what can I do in this particular case? I can say that this is equal to a to the power of x times... Now, instead of a to the power of delta x, I will have e to the power of delta x times logarithm a minus 1 divided by... Now, you see, I would like to have in the denominator this, because then if delta x goes to 0, delta x times logarithm a also goes to 0. And then this thing would go to 1. So to compensate this, I have to multiply it by natural logarithm a, right? And now you see that this is 1, right? In the limit. So what's remaining is this. So my derivative is equal to logarithm a times a to the power of x. This is the derivative of this function. And natural logarithm is logarithm with a base e. The same e, very important number in all mathematics. Alright, so these are two very simple derivatives from exponential functions. Well, next would be probably trigonometric. That would be a separate discussion. Thanks very much and thanks and good luck.