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# Basic derivative rules (Part 2)

## Video transcript

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 use the definition of limits to feel good about that now 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 then to feel good that they're actually true so 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 and once again maybe we could make actually we could make a graphical argument for why that is true it's going this is going to multiply the 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 a what 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 whoops my pen doesn't work F prime of X is going to be equal to the limit as X as H approaches 0 I'm using that definition of f of X plus h minus f of X all of that over all of that over H well what is f of X plus h so this is the limit as H approaches 0 f of X plus h 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 and 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 is I factored 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 and 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 and I will 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 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 let's say I don't know J of X I don't know J oh yes sure why not J back you don't see a lot of J of X is 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 been true if this instead of being a positive here if this was a negative or this instead of addition if this was subtraction if you're the sum or difference of two functions and your derivative is going to be the sum or the difference of their derivatives and 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 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 we're going to 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 can 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 going to 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 I could write it like 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 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 J prime of X and we're done and 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 of themselves are somewhat straightforward you could probably guess that at them but it's nice to use the definition of our derivatives to actually feel feel that they are very good conclusions to make