 Hi, I'm Zor. Welcome to InUzor Education. We continue talking about derivatives as a course of advanced mathematics for teenagers. It's presented on Unizor.com. I suggest you to watch this lecture from this particular website because it has nice comments, actually very detailed notes for each lecture. And there is certain functionality which you might be interested in, like taking exams for instance, etc. Now, today we will talk about properties of derivatives. In particular, how to take derivative of a function which is a linear combination of our other functions. So let's consider you have a function h of x which is a linear combination of two other functions, f at x and g at x. And my purpose right now is to have a derivative of function h in terms of derivatives of function f and g. Well, first of all, let's talk about domain. Since we are talking about sum of two functions, we have to have certain domain where both functions are defined. So I assume that there is certain domain where both functions are defined. Now, since I'm talking about derivatives, I assume that this domain includes all the points where these two functions are differentiable. So their derivatives exist. So this is not only the main where the functions are defined, but this is the main where both functions have derivative. So both are differentiable on this particular domain. Now, here I can talk about function h, which is also defined on this particular domain. And my statement is that the derivative of the function h is the same linear combination of derivatives of functions f and g. Exactly, the same linear combinations which has the same coefficients as original. So this is basically a theorem which I'm going to prove. Well, it's extremely easy thing and it doesn't really deserve maybe such a great name as a theorem. But anyway, it's a theorem. It's a statement which I'm going to prove. And the proof is actually very easy. It's all based on the similar property of the limits. Limit of linear combination is linear combination of limits. So let's just do it step by step very easily. Now, if my argument x belongs to interval a, b, and I have an increment, then my function also will have increment f of x plus delta x minus f of x, which is delta f of x, and also g would have exactly similar delta g of x. Similarly, my function h of x would also have an increment. Now, what is this particular increment is equal to? Let's call it delta h of x. Okay, it's function of x plus delta x, which is a times f of x plus delta x plus bg of x plus delta x. This is what h of x plus delta x is. It's a times f of this argument plus b times g of this argument. Minus h of x, so it's minus a f of x minus b g of x. Now, obviously, if I will regroup it, this minus this gives me this, and this minus this gives me this. So it's equal to a delta f of x plus b delta g of x. Now it's time to go to a limit. The derivative of the function h of x is equal to limit of delta h of x divided by delta x as delta x tends to zero. So h of x plus delta x minus h of x I have replaced with delta h of x. Now, as we know, I have expressed delta h of x in this term. So it's limit of a delta f of x divided by delta x plus b delta g of x divided by delta x. Now, this is a limit of linear combination of two variables which do have their limit. The limit of this one is, as we know, derivative of f, and this is derivative of g. So that's why this is equal to a f of x plus g derivative of x. So it's all based on the corresponding property of the limits. The limit of the linear combination of variables, each of which has a limit, we have assumed that both f of x and g of x are differentiable on this domain. So that's why this has a limit and this has a limit. That's why the limit of their linear combination is equal to linear combination of the limits. And this is the end of the proof. The derivative of the linear combination of two functions is linear combination of their derivatives. End of story. Now a couple of examples which would help you to use this particular property if you would like to take a derivative of some a little bit more complex function than we did it before. For instance, 5 sine of x minus 7 cosine of x. And I would like to take derivative of this. Well, again, this is linear combination of the functions. So we know that the derivative of linear combination is linear combination of derivative. 5 derivative of sine which is cosine minus 7 the derivative of cosine which is minus sine. So it's plus sine of x. That's it. Next, I'll use different notation d of 2x square minus 3x is equal to... So this is just a different notation of the derivative. So it's linear combination of two functions. 2 times x square and minus 3 times x. And we know that the derivative of the x to the power of n is nx to the power of n minus 1, right? Remember that. So that's why this is equal to 2 times derivative of x square which is 2x minus 3 derivative of x to the first degree, first power. So it's 1 times x to the power of 0 which is 1. So it's 4x minus 3. Next, d of dx to e to the power of x minus x. Again, linear combination of two functions. So it's 2 times derivative of e to the power of x which is e to the power of x minus derivative of x which is 1. x to the power of 1 derivative would be 1 times x to the power of 0 which is 1. So it's 1. And the last one I'll use yet another different notation for the derivative. Used much rarer, by the way. And I will do a combination of the polynomial, exponential and trigonometric function. Well, now I have proved this theorem for two components. Now, obviously you can prove it for 3 or n or any number of components just basically dividing it into corresponding groups, right? So if I have a, f, b, g times c times h, let's say fx. Well, you group first this and then separate the third component and use your theorem for two components only. And then derivative of this group will be in turn the same kind of linear combination of derivative. So, obviously I can expand this to any number of components, linear combination of the functions. If you want to derive, if you want to take a derivative it would be linear combination of derivatives. So it's linear combination of these derivatives which is derivative of 2 to the power of x square which is 2 times fx square. It's 2x minus 3e to the power of x. It's minus 3 and derivative of e to the power of x is e to the power of x. 4 cosine is minus 4 sin x. So that's the result. Well, obviously 2 times 2 you can change to 4. So that's the result. These are just examples of how you use this very simple and obvious property of the derivative to basically take derivative of a rather complex expression like this one, for instance. And there are other properties which will be addressed in the next lecture. That's it for a while. Thank you very much and good luck.