 Hi, I'm Zor. Welcome to Indizor Education. Today we will talk about derivatives of higher order, like second derivative, third derivative, etc. Well, first of all, let me just explain the concept behind this. What is the derivative? The definition of the derivative of the function f of x, which is defined at certain interval, which might be infinite in both sides. At point x0 is a limit of increment of the function. At point x0, given the argument increment of delta x, as this increment of the argument tends to 0, if this limit exists, obviously. So, let's just think about what this represents. We have to have the function and we have to have the point. And at every point where this limit exists, we have the definition of the derivative. So, basically, what I'm saying is that this whole thing represents a function, which obviously depends on the original function f, but it's defined at point x0. Now, let's consider this function and consider all these argument points within this interval, where this limit exists. So, for all these points x0, where this limit exists, function g of x0 is defined. So, basically, it's a function. And what's the domain of this function? Well, it's not greater than this, obviously, domain, because it's only for those x0 from this domain, where this limit might exist. But it might exist on a subset, on a smaller set of points, smaller than this interval, if there are certain points where this limit does not exist. So, in any case, this function is defined on some kind of a domain, which is a subset, or maybe exactly equal to AB. And it has certain values. The value is this particular limit. So, the derivative is a function, which is defined on maybe narrower, but maybe exactly the same interval as the original function f at x is defined. And obviously, it depends on the function itself. And traditionally, this is denoted as f and prime or something. Another definition, another notation is this. But now, it doesn't really make sense to use x0, because we're talking not about a particular point where this particular limit exists. We can talk about any point from this interval where this limit exists. And I can actually just scratch this 0 and talk about new function f prime of x, or df at x divided by dx, which denotes my derivative of the function f at point x from this interval. Obviously, with a node where this limit exists. Okay, now, remember a few samples which we have, like the derivative of a sine is a cosine, etc. It's a function. So, if this is a function, I can actually consider a second derivative, which is a derivative of this derivative. How is it defined? Now, we don't need this g anymore. Well, it's denoted as f second of x, which is limit f prime of x minus f plus delta x minus f prime of x divided by f, again, wrong, divided by delta x. Because delta x goes to zero. Again, obviously, if this limit exists. So, first requirement is that the derivative exists, which means this limit should exist at point x. And then, if the derivative of this function, which is this limit, exists at point x, then we have a second derivative. So, it might be defined on even narrower area. So, basically, if you forget about this if the limit exists node, which I'm trying to emphasize every time. Basically, we're talking about one function defined on some axis, which is dependent on original function. Then another function, which is dependent on this function. And we can continue this as long as these limits exist. We can continue this process of building the second, the third, the fourth derivative. It's a derivative of the derivative of the derivative as many times as you want as long as these limits exist. Well, sometimes there are cases, which you can infinite number of derivatives take. Derivative from derivative, infinite number. If this limit is defined, for instance, for every point and throughout the whole process, then here you go. Now, the third derivative is obviously f. It's limit of f second derivative as delta x goes to zero. And then, obviously, you can define f four. I'm using the Roman numerals, which is the derivative of the third derivative, f five, etc. So, usually, with these higher order derivatives, derivatives from derivatives, we can use Roman numerals. Another notation is, let me talk about the second derivative, this one. d square x d f at x. I'm sorry. Let me just have a little bit more real estate here. d second f at x divided by d x square. So, that's another notation for the second derivative of function f. So, this is one with second slashes or whatever you call it, Roman numeral. And this is another notation. And the third one and the fourth one, etc. So, these are regular integer numbers. d cube f at x divided by d x cube. That's the third derivative. d fourth, etc. So, these are the concepts behind taking the second, the third, the fourth derivatives and any other higher order derivatives. So, that's what they are. That's what higher order derivatives are. It's derivative from the derivative. And now, let's just have a couple of examples. And in my examples, functions have this limit everywhere. So, the first function is f at x is equal to a constant. So, the graph is, obviously, if this is a, so the graph is straight line. Okay. What's the derivative of the constant a? Well, we already actually spoke about this before. But in any case, I mean, direct calculation f of x plus delta x minus f at x divided by delta x is always equals to zero because f at x plus delta x is a and f at x is also a, so it's zero. So, the limit is zero. So, the first derivative is zero. Now, zero is also a constant, which means that the second derivative, the constant, the first derivative is the constant. So, the second derivative would be derivative from the constant. And we know this is zero again. And the third one and the fourth one up to infinity. So, this function is differentiable up to any degree. So, any order, the second, the third, the fourth order derivatives, they all exist. And starting from the first, they're all equal to zero. So, that's my first very, very trivial example. Now, how about function equals x to the power of n? Well, before, we actually made some calculations and we found that the first derivative is equal to n x to the n minus one, right? Now, it's very easy to prove, and I will probably talk about this separately, that if you have a multiplier, a factor times some function, then this multiplier can be taken out from the differentiation. Why? Primarily because, again, remember, limit f at x plus delta x minus f at x divided by delta x. If f at x can be represented as some multiplier times something, this multiplier will be here and there. So, it will be outside of the limit, which means that the derivative of the function of this type can be represented as the same factor times derivative of this. Now, what is derivative of x to the power of n minus one? Well, I mean, we know about this. So, obviously, this would be this times this, right? So, the power becomes a multiplier and the power is reduced by one always. So, if it's already n minus one, so it will be n minus one and this will be n minus two. So, what will be the third derivative? n n minus one and minus two x to the power of n minus three. How many times it will last? Well, it will last actually n times. So, you see, the third is minus three. So, the nth, I'll use n in parenthesis as an indicator that this is nth derivative, would be, well, obviously, it will be n factorial, right? n times n minus one, et cetera, times one. And here you will have x to the power of zero, right? n minus n, which is one. Now, this is a constant, right? So, the next one would be zero because this is now a constant. So, nth different derivatives as nth and after n starting from like n plus one, you will have zero, zero, zero, zero, zero as derivatives. Also, infinitely times differentiable, differentiable function, infinitely differentiable. Okay, next example. Next example is a to the power of x, where a is some kind of positive constant. Now, the first derivative we already log a rhythm of a times a to the power of x. So, we already calculated this in the previous lecture. So, what is the second derivative? Well, I was talking before that the multiplier can be just factored out, and then the derivative of the rest can be taken. Which means we will have logarithm of a and then again logarithm of a. So, it will be square. The third derivative would be cube. The nth derivative would be logarithm to the power of n of a and a to the power of x. And by the way, if my a is equal to e, remember what e is, right? Then this natural logarithm of a is actually equal to one, right? Because this is natural logarithm is logarithm with a base e. So, which power I should raise e to get e? Obviously, one. So, it will be one, one, one, one, one to the power of n is obviously the same one. So, e to the power of x is e to the power of x. So, any derivative from the function e to the power of x, the first derivative, the second derivative, the nth derivative, any derivative would be exactly the same as this function. And this is absolutely remarkable property of the function e to the power of x. That's why it's so special. I mean, all these exponential growth, etc., which we see observed in the nature, I mean, it's really very much related to a very special property of this function. It's a very important function e to the power of x. Okay, my last example is from trigonometry. If my function is equal to sin of x, okay, we already spoke about the first derivative is equal to cos sin of x. Now, the second derivative is minus sin of x, right? The derivative from the cosine. Already spoke about this. Now, the third derivative is minus, because this is a multiplier, right? And then derivative of sin, which is cosine. Fourth derivative is again minus, and derivative from the cosine, which is minus sin, which is minus and minus will be positive sin. So, we see that the fourth derivative is exactly the same as the original function. And then it repeats itself. The fifth derivative would be like the first, the sixth like the second, etc. So, it's periodically actually repeats itself. All right, these are just simple examples of different higher order derivatives of certain basic functions. And in reality, obviously, the functions which you will be dealing with will be a combination of these more or less. All right, so that's it for this particular item. So, you know what higher order derivatives actually is. I do recommend you to take a look at the notes for this lecture. They're very brief. But just to make sure that you basically inculcate this concept into your mind, it's important to read it on Unisor.com. That's it. Thank you very much and good luck.