Applying First Principles to x² (1 of 2: Finding the Derivative)

picking up where we left off from yesterday what we finished with was thinking about rise over run in the specific context where you have a curve like this and it's great is tuned all the time right so gradient like we usually with straight lines who want to jump in cylinder we say Oh gradient we'll just call it in right because it's just a number it's just a constant it's not changing a small variable right but here M is not gonna cut it it's not gonna cut because in fact the gradient is not a number smaller constant it is a function itself but greedy is changing wherever you look right so therefore we replace this idea of the gradient as a constant with the gradient as a function in fact we call it the gradient function right and rather than say rise over run which is just kind of a nice and then want equate ever and bring what crazy news we said rise is really the change D Delta the change in Y and not it is really the change in X and that's where we get this dy over DX notation from ok now one word I didn't introduce yesterday is that we call this thing not just the gradient right but now that it's a function it's it's own thing we call it the derivative and derivative you know it just means it comes from something else right namely it comes from the actual function right we let this way of taking this rise over run and putting on all these different points and comparing out with limits what was happening in a particular point on there ok so this is the introduction actual thing that we said was fit first principles right we said f dash which is just yet another way of indicating the TEC notation right is equal to the limit as H approaches you were off now what was that numerator what was it f off f of X plus h and then we take away f of X now just remember why do we do that it's because f of X plus h is here and f of X is here we just want the difference all it is that's the rise and of course there's that run do you remember when I said to you in traditions topic the worst thing that you can do is just get get to work memorizing stuff and not know where it comes from I remember sitting next to you when I was in year 11 sitting next to people who just like look it's just just memorize it okay doesn't matter what any of it means okay that's the quickest way to get on the path of not actually understanding what's happening this is rise right this is run one last thing what difference does that make if it weren't there that limit as H approaches zero if it weren't there it's not meaningless but what would it be it be something else Mary it had been the gradient of not that another tangent the tangent all we want right but if you've got two points that are actually a part it'll be the secant wantnt right it's not meaningless until we started it's just not what we want this is the grain of the tangent the derivative without that and many people like I have to write this over and over again without that it's the grating of the second we're not after that okay so now that's what we stopped right now I want to apply it to a particular situation so here's a function right now the one we'll start with that you just want to start with is just the problem x squared if you see f of X right then you can use this F dash X notation okay if on the other hand all you see is this there is no function called F right so none of this really makes sense f dash what does any of that in here you would have to use this notation up here right if you've got wise you just do one of the X if you've got F's you use F and F dash okay now I'm just going to come back because for this example we're just on I do actually want to go with a score to make this a bit easier so this is f of X okay let's proceed through this right if f of X is that then f dash X is going to equal 2 now it begins with justice substitution if we know what F is I should know what all those pieces are in there okay so we'll begin with the limit because I'm interested in the tangent and not the secant if f of X is x squared then f of X plus h is X plus h all swear you agreed that and there's f of X just as we defined it we're dividing through by H okay now what we're about to do is what we call evaluating the derivative evaluated derivative from first principles this is our starting point okay what we're trying to do is manipulate this in such a way such an I can actually put H equals zero in there and see what happens I can't do that right now because H is on the denominator you divide by zero it explodes okay so I want to reshape this into something that gets rid of H being just by itself on the denominator so let's give this a go obviously you look at the numerator that's the only thing you can do anything with here the numerator the denominator is just super fine so I write my limit because I want the tangent not the secret you're gonna get really sick of me saying that but I want you to get in your head it's really important to keep on saying on the numerator perfect square when you expand it you get x squared plus two HX plus h squared and then there's that minus x squared tacking along the end okay and then of course everything is divided by H right just because you're not doing any work with them don't forget to write this or the denominator they're still there and they're still critically important having expanded you can see I can do something with this now can I go next squared on the front minus x squared on the back and they're going to each other out okay now the reason why this is good is because now I have a common factor of H on the numerator right common factor H let me take it out we factorize it limit as H approaches your because I've want the tangent not the secret oh what's on the numerator H outside off-page good we're still at this still in H pain around there that's all divided into three badge okay fantastic now I can cancel denominators go on yep canceling out yes this is exactly right I'm glad you raised that point why can I get rid of this okay let me pull back to you like this this okay now what's this thing a little plot okay sorry yes now I also ordered plug for say um what does this thing look like even though it's gonna make squared on the top because of what you get on the bottom you're actually gonna get a straight line aren't you can you factorize X minus one you can't with that so what does this thing actually look like it looks like this straight line with one difference namely there's a hole there it's just a hole right so X minus 1 here and X equals the negative 1 is somewhere in here I guess this method will hold it okay so our problem is I can't simply input x equals negative 1 because then as you notice I am multiplying by here and divided by zero and it loses me okay however what I'm just come back to when we define this idea of a limit right what I try to think about is what am I getting to was right and I'm actually getting towards something real even though I can't be there itself any more than I can calculate the gradient of you know a point to itself that's what this is really doing rise over run as the two points get close to get on right I couldn't actually calculate that but I can still see what it's approaching and if it's approaching the same thing from both angles then that's what that's great I can take that as a meaningful result okay so though a very good point to mention because it's like yeah why can I do that and the answer is because it can't be actually equal to zero so I can take it out into the equation expression I say now that I've done this this is the last time I'm going to write the limit as H approaches zero because I want say it with me the tangent not the secret right just like a hammer okay so now that I'm there I've gotten rid of H on the denominator right oh by the way H can be on the denominator just not by itself because when it's by itself the denominator becomes zero as you say Charlie now I actually can say well let's just see what happens when H is zero and the answer is it's 2x plus zero don't miss the plus zero it's not trivial okay just like multiply my one sometimes it's not trivial adding zero is not trivial because I see it comes from here and that of course is just 2x okay what's chain

Channel: Eddie Woo

Views: 336,153

Rating: 4.9496179 out of 5

Keywords: math, maths, mathematics

Id: fXYhyyJpFe8

Length: 9min 31sec (571 seconds)

Published: Sun Jun 28 2015