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# Product rule proof

## Video transcript

what I hope to do in this video is give you a satisfying proof of the product rule so let's just go let's just start with our definition of a derivative so if I have the function f of X and if I wanted to take the derivative of it by definition by definition the derivative of f of X is the limit as H approaches 0 of f of X plus h minus f of X all of that over all of that over H if we want to think of visually this is the slope of the tangent line and all of that but now I want to do something a little bit more interesting I want to find the derivative with respect to X not just of f of X but the product of two functions f of X times G of X and if I can come up with a simple thing for this that essentially is the product rule well if we just apply the definition of a derivative that means I'm going to take the limit as H approaches 0 in the denominator I'm going to have an H in the denominator will write a big it's going to be a big rational expression in the denominator I'm going to have an H and then we'll evaluate this thing at X plus h so that's going to be f of X plus h G of X plus h and from that I'm going to subtract this thing evaluated f of X or sorry this thing evaluated X so that's going to be f of X times G of X and I'm going to put a big awkward space here and you're going to see why in a second so if I just if I evaluate this at X there's just going to be minus f of X G of X all I did so far is I just applied the definition of the derivative instead of applying it to f of X I applied it to f of X times G of X so you have f of X plus h G of X plus h minus f of X G of X all of that over H limit as H approaches 0 now why did I put this big awkward space here because just the way I've written it right now it doesn't seem easy to algebraically manipulate I don't know how to evaluate this limit there doesn't seem to be anything obvious to do and what I'm about to show you I guess you could view it as a little bit of a trick I can't claim that I would have figured it out on my own maybe eventually if I were spending hours on it and I'm assuming somebody was fumbling with it long enough they said oh wait wait look if I just add and subtract the same term here I can begin to algebraically manipulate it and get it to what we all know is the classic product rule so what do I add and subtract here well let me give you a clue so if we have plus actually let me change this minus f of X plus h G of X I can't just subtract if I subtract it I've got to add it to so I don't change the value of this expression so plus f of X plus h G of X now I haven't changed the value I just added and subtracted the same thing but now this thing can be manipulated in interesting algebraic ways to get us to what we all love about the product rule and at any point you get inspired I encourage you to pause this video well to keep going let's just let's just keep exploring this this expression so all of this is going to be equal to it's all going to be equal to the limit as H approaches 0 so the first thing I'm going to do is I'm going to look at I'm going to look at this part this part of the expression and in particular let's see I am going to factor out an f of X plus h so if you factor out an f of X plus h this part right over here is going to be f of X plus h f of X plus h times you're going to be left with G of X plus h G of the slightly shader different shade of green G of X plus h that's that there minus G of X minus G of X whoops I forgot the parentheses oh it's a different color I got a new software program and it's making it hard for me to change colors my apologies this it's not a straightforward proof and the least I could do is change colors smoothly alright G of X plus h minus G of X that's that one right over there and then all of that over this H all of that over H so that's this part here and then this part over here this part over here and actually it's still over H so let me actually circle it like this so this part over here I can write as so then we're going to have plus actually here let me let me let me let me factor out a G of X here so plus G of X plus G of x times this f of X plus h times f of X plus h minus this f of X minus that f of X all of that over H all of that over H now we know from our limit properties the limit of all of this business well that's just going to be the same thing as the limit of this as X as H approaches zero plus the limit of this as H approaches zero and then the limit of the product is going to be the same thing as a product of the limits so if I use both of those limit properties I can rewrite this whole thing as the limit let me give myself some real estate the limit as H approaches zero of f of X plus h of f of X plus h x times the limit as H approaches zero of all of this business G of X plus h minus G of X minus G of X all of that over H I think you might see where this is going very exciting all right plus a plus the limit let me write a little bit more clearly plus the limit as H approaches 0 of G of X are nice brown color G of X times now remember product here the limit the limit as H approaches zero of f of X plus h of f of X plus h minus f of X minus f of X all of that all of that over H and let me put the parentheses where they're appropriate so that that that that and all I did here the limit the limit of this sum that's going to be the sum of the limits that's going to be the limit of this plus the limit of that and then the limit of the products is going to be the same thing as a product of the limits so I just use those limit properties here but now let's evaluate them what's the limit and I'll do it in different colors what's this thing right over here the limit as H approaches zero of f of X plus h well that's just going to be f of X now this is the exciting part what is this the limit as H approaches 0 of G of X plus h minus G of X over H well let's just start to that's the definition of our derivative that's the derivative of G so this is going to be this is going to be the derivative of G of X which is going to be G prime of X G prime of X so you're multiplying these two and then you're going to have plus well still limit as H approaches 0 is G of X well there's not even any H in here so this is just going to be G of X so plus G of X times the limit so let's see this one is in brown and the last one I'll do in yellow times the limit as H approaches 0 and we're getting very close to serve the drum roll should be starting limit as H approaches 0 of f of X plus h minus f of X over H well that's the definition of the derivative of f of X this is f prime of x times F prime of X so there you have it the derivative of f of X times G of X is this and if I wanted to write in a little bit more condensed form it is equal to it is equal to f of X time's the derivative of G with respect to X times the derivative of G with respect to X plus G of X plus G of x times the derivative of F with respect to X F with respect to X or another way to think about it this is it's the first function times the derivative of the second plus the second function times the derivative of the first this is the proof or a proof there's actually others of the product rule
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