Proofs of the constant multiple and sum/difference derivative rules

- [Voiceover] In the last video we introduced you to the derivative property right over here that if my function is equal to some constant that the derivative is going to be zero at any X, and we made a graphical argument and we also used the definition of the limits to feel good about that. Now let's give a few more of these properties and these are core properties as you throughout the rest of your calculus life your career, you will be using some combination of these properties to find derivatives. So it's good to, one, know about them, and to feel good that they're actually true. The second one is if my function F of X is equal to some constant times another function G of X well, then the derivative of F of X is going to be equal to that same constant. Let me do it, that same constant times the derivative of the G of X. And once again, maybe we can make a actually we could make a graphical argument for why that is true. This is going to multiply the slope is one way to think about it. But it's easier to make an algebraic argument just using, frankly we can use either one of these definitions of the derivative. I'll use the one on the right because it feels more general although you can say, well this is true for A, but X could be any A. But I'll just use the one on the right. So, if we want to find F prime of X, F prime of X using this definition, we know, we know. F prime of X is going to be equal to the limit as X, I'm sorry, as H approaches zero. I'm using that definition. Of F of X plus H minus F of X all of that over H. Well, what is F of X plus H? This is the limit as H approaches zero. F of X plus H is K times G of X plus H. Minus F of X, well that's just KG of X. KG of X. All of that over H. And then you can factor the K out. This is going to be equal to the limit as H approaches zero of K times G of X plus H minus G of X all of that over H. All I did was factor that K out. And we know from our limit properties that this is the same thing as K times the limit as H approaches zero of G of X plus H minus G of X, all of that over H. And of course, all of this business right over there. That is just G prime of X. So, this is equal to K times G prime of X. And I know what you might be thinking. Whoa, hey this feels like it was probably going to be true, so I just assumed it was true. But you can't just assume that. I don't know, sometimes you can when you're first trying to get your head around it you can say, oh, this seems like a reasonable thing. But in math, we like to really know that it is true otherwise we will build all sorts of conclusions based on unsound foundations. This allows us to ensure that this is something that we can do. So, this is, it's good to go through what might feel like a little bit of work to get to this conclusion. Now let's do the third property. The third property is the idea that if I have some function that's the sum or the difference of two other functions, so G of X, and let's see, I'm using H a lot so, H, let's say, I don't know, J of X. J, yeah sure, why not, J of X. You don't see a lot of J of X's out there. Well then, well then, F prime of X is going to be equal to G prime of X plus J prime of X. And this would also have been true if this is, instead of it being a positive here, if this was a negative. Instead of addition, if this was subtraction. If the sum were a difference of two functions then your derivative is gonna be the sum or the difference of their derivatives. And once again, we can just go to the definition of F prime of X. So, F prime of X is going to be equal to the limit as H approaches zero of F of X plus H. But what is F of X plus H? Well, that's G of X plus H plus J of X plus H. So that's F of X plus H minus F of X. So F of X is G of X. G of X plus J of X plus J of X. Notice, this is F of X plus H minus F of X and we're gonna put all of that over H. So we can put all of that all of that over H. Well, what is that equal to? Well, we could just rearrange what we see on top here. This is equal to the limit as H approaches zero. Well let's see, all the mentions of G of X I'm gonna put up front. So G of X plus H minus G of X. Plus J of X plus H minus J of X. And then all of that, I could write it as this, all of that over H. Or, I could, that's the same thing as this over H plus that over H. And once again, we know from our limit properties that that is the exact same thing as the limit as H approaches zero, of G of X plus H minus G of X all of that over H plus the limit as H approaches zero of J of X plus H minus J of X all of that over H. And this, right over here, that is the definition of G prime of X. And this right over here is J prime, J prime of X and we're done. And this, instead of a positive, this was instead of addition this was subtraction. Well then that subtraction would carry through and then instead of addition here we would have subtraction. So hopefully this makes you feel good about these properties. The properties themselves are somewhat straightforward. You could probably guess at them. But it's nice to use the definition of our derivatives to actually feel that they are very good conclusions to make.