Basic derivative rules (Part 2) | Derivative rules | AP Calculus AB | Khan Academy

In the last video, we introduced you to the derivative property right over here: that if my function is equal to some constant, then the derivative is going to be zero at any X. We made a graphical argument, and we also used the definition of limits to feel good about that. Now, let's give a few more of these properties, and these are core properties. As you go throughout the rest of your calculus life and career, you will be using some combination of these properties to find derivatives. So it's good to know about them and then 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 G of X. Once again, 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 could use either one of these definitions for the derivative. I'll use the one on the right because it feels more general. Although you could say, well this is true for any X, I’ll just use the one on the right.

So if we want to find F prime of X, using this definition, we know—oops, my pen doesn't work! F prime of X is going to be equal to the limit as H approaches 0 of f of X plus h minus f of X, all of that over H. Well, what is f of X plus h? So this is the limit as H approaches 0 of f of X plus h, which is K times G of X plus h minus f of X. Well, that's just K G of X. K G of X, all of that over H.

Then you can factor the K out. This is going to be equal to the limit as H approaches 0 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 0 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.

I know what you might be thinking: well 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 will say, sometimes you can, you know when you're first trying to get your head around it, you can tell how 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, look, this is something that we can do. So 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 difference of two other functions G of X and, let’s see, I'm using H a lot. So, let’s say, I don't know, J of X—I don’t know, J. Oh yes, sure, why not, J. You don’t see a lot of J of X's out there. Well then, F prime of X is going to be equal to G prime of X plus J prime of X. This would also have been true if instead of being a positive here, this was a negative, or if this instead of addition, if this was subtraction. If it’s the sum or difference of two functions, then your derivative is going to be the sum or the difference of their derivatives.

Once again, we can just go to the limit, the definition of f prime of X. So, f prime of X is going to be equal to the limit as H approaches 0 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 plus J of X. Notice this is f of X plus h minus f of X. We're going to put all of that over H.

So we can put all of that over H. Well, what is that equal to? Well, we can just rearrange what we see on top here. This is equal to the limit as H approaches 0. Well, let’s see all the mentions of G of X. I'm going to put up front: 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 like this: all of that over H or I could—that's the same thing as this over H plus that over H.

Once again, we know from our limit properties that that is the exact same thing as the limit as H approaches 0 of G of X plus h minus G of X, all of that over H plus the limit as H approaches 0 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 of X. And we're done!

If this instead of a positive, if this was instead of addition, if 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.