what's the gradient at that point because there are no two points to compare by definition so here's what they both did they said imagine a circle okay we can picture a tangent along here in fact that's where our language of tangents first comes from okay now it's hard to do the maths with this thing because you just got that one point ah but if I can get towards this tangent and think of it in terms of something else that I can calculate something which does have two points then I can work with this for example if I put in a whole bunch of parallel lines here okay all of these guys these 1 2 3 4 that I've just drawn what are they called they intersect the circle twice they're called SE these guys right these guys do have two points so I'm just going to think of the circle because that's where your secant and tangent language comes from and then we'll come over to a graph in a second can you see the length of a chord that's cut off by this sein right the length of this cord as I get closer and closer to becoming a tangent as I move this way right the length of that chord is getting smaller you agree right in fact eventually once you get to the tangent the length of that chord the length of that chord is zero is it not it's it's it's not actually a cord it's just a single point that's why it's actually zero right so what Newton wanted to do was take this problem apart by thinking of it as one of these problems he thought what if what if I could I can understand the behavior of this SE what if I could imagine it as it approaches a chord of length zero if it approaches a chord of length zero so both of these guys both of these guys said I need some language I need some language to talk about things that approach something which you know I can't actually get to but I can still do the calculations on it so they said let's talk about limits they introduced this notation something getting closer and closer and closer until you're actually there right you make conclusions off of off of things that you know so that you can draw some information about something you don't know okay so now here's where the rubber hits the road draw for me a new um new set of axes let's just go first coordinate that's all you need now draw for me a curve draw for me a curve and let's call this some function of X some function of X okay now if what I'm after is the gradient of a tangent gradient of a tangent okay what I'm going to start thinking about is the gradient of a secant and think about what happens to that as it gets closer and closer to becoming a tangent right so in order to get a secant that's not hard you just pick any two points you like like here and here okay and can you see like if the actual point after is like somewhere in the middle here okay if you were to draw that tangent it's actually not that far off in terms of its gradient compared to this secet right like I'm in the right ballpark yeah so I've got these two points got these two points let's get some values on this all right if I imagine this is some x coordinate who cares what it is I'll just call it X I want it to be anywhere that I like I want to be able to input some value there okay and what I'll do is I'll think about going from that point and going a bit further right going like a distance of say they call it h okay you can think that H is kind of like for height but that's not that's not the best description cuz it's further anyway H is the convention there are other conventions as well sometimes you'll see it called Delta X we'll talk about Y in a second now if that little distance there it's just a small distance is H then what's the coordinate of this point over here or I should say the X X Plus right it's just that plus that that gets you over there okay so now if I want to work out the gradient of this secant here right the gradient of the secant I've got x coordinates I'm going to need Y coordinates I know right let's come over to here and here okay now I don't know what this function is it could be anything it's F ofx right so if this x coordinate is X just X right what am I going to be up here f of x good so for example if this were If This Were X2 right Y2 then that would be X and this would be X2 right simple and this is X Plus H so my corresponding y value over here will be f of x plus h whatever F happens to be okay so now what have I got here what have I got I've got what I'm working out is the gradient of the SE right the gradient of this guy here okay and it's literally just rise over run it's just rise over run okay so what's my rise okay it's the distance between these two points these two y values is it not okay what is that distance this take away this yeah that's all that's all that vertical distance is okay so rise I've got that okay now run is the horizontal distance that I'm going that corresponds to that which of course I've defined as just H it's just H so if I have some value of x wherever you like okay and I have some function you define it to be whatever you like okay this will give me the gradi of a secant that's anywhere you want okay but I don't want a secant really what I really want is at a single point where this distance in here I don't want it to be this big gaping Gap in here okay I want it to close in I want it to come together just like these guys are coming together right so what I want is this guy here H I want it to get very very very small does that make sense now I already have language for this right when it becomes zero if what's I don't know what's happening I'm going to get not a secret not a see but a tangent do you agree with that right that's what I want what I want is for H to be zero now H can't really be zero cuz look I have to divide by H I can't really be zero but I can think about what happens as I get closer to zero I just have to say tell me what the limit is right tell me what the limit is just like we looked at before if I gave you a trivial example okay we looked at U we looked at evaluating limits like that okay now that function there that I put over here it's got a ho at five doesn't it there's you can't equal five because it's on the denominator so xal 5 is not going to work okay but all I need to do is I just have to say well that top thing what can you do to that top thing I can factorize it it's the difference of squares right so it's going to be x + 5 x - 5 and because of that our denominator goes no problem okay so at that point I can say oh that thing that thing that approaches five and I can actually put in five so of course it's just going to be 10 right so even though this original thing I can't put five in I can see what happens as it approaches five it's going to go there okay from above and below you can go ahead and try it out okay so here I can't put in zero but I can think about what happens as I get ever closer to it okay so this thing here this guy is a big deal okay this was Live's and Newton's big watershed moment they're like I can't I can't work out by just putting in zero right at least into this which is what it is by definition rise over one but I can still do maths with it I can still do maths with it even if we can't actually put zero in this is what we call the first principles of calculus the first principles of calculus what are we actually working out we're working out the gradient of the tangent okay but this started to get a bit awkward in terms of like talking about all this stuff so they introduced new language and they introduced um new notation okay so the new language they introduced was they called this thing right like I could get it at a particular point I get at a particular point but I could get it at any point I like right and it's going to change all the way so rather than this thing don't which has a gradient okay this thing over here is going to be a function it changes right so it's actually called the gradient function because the gradient is going to take on different values depending on where you look depending on what value of x you look at okay now in the same way he said look this rise run business rise over run it's not working because there's not really a run it's not really a run okay so they said look this is just the change in y divided by the change in X that's that's all that's really happening right so you guys know the symbol in science right for change is a Delta so use a d for Delta and they say the change in y That's y's right and you compare it you get it ratio with its change in X that's the Run except run is an inadequate word cuz there's not really any run happening here right so Dy change in y over DX change in X it means rise over run but in that case where the rise and run are both tending towards zero cuz you're looking at this infinitesimally small spot the rise is going to be zero the run's going to be zero but you can still see how it behaves