If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Formal definition of the derivative as a limit

AP.CALC:
CHA‑2 (EU)
,
CHA‑2.B (LO)
,
CHA‑2.B.2 (EK)
,
CHA‑2.B.3 (EK)
,
CHA‑2.B.4 (EK)

## Video transcript

we're first exposed to the idea of a slope of a line that early on in our algebra careers but I figure it never hurts to review it a bit so let me draw some axes that is my y-axis maybe I should call it my f of X axis Y is equal to f of X and let me draw my x-axis just like that that is my x axis and let me draw a line so let me draw a line like this and what we want to do is remind ourselves how do we find the slope how do we find the slope of that line and what we do is we take two points on the line we take two points on the line so let's say we take this point right here let's say that that is the point X is equal to a and then what would this be this would be the point f of a where the function is going to be some line we could write f of X is going to be equal to MX plus B we don't know what m and B are but this is all a little bit of review so this is a this is and then the Y value is what happens to the function when you evaluate it at a so that's that point right there and then we could take another point on this line let's say we take the point B right there and then this coordinate up here is going to be the point B be f of B right because this is just the point when you evaluate the function at B you put B in here you're going to get that point right there so let me just draw a little line right there so that is f of B right there actually let me make it clear that this coordinate right here is the point a F of a so how do we find the slope between these two points or it more generally of this entire line because a line has the holes the slope is consistent the whole way through it and we know that once we find the slope that's actually going to be the value of this AB that's all a review of your algebra but how do we do it well a couple of ways to think about it slope slope is equal to rise over run that you might have seen that when you first learn algebra or another way of writing it it's change in Y over change in X so let's figure out what the change in Y over the change in X is for this particular case so the change in Y is equal to what well let's just take you can you could take this guy as being the first point or that guy being the first point but let's take since this guy has a larger X and a larger Y let's start with him so the change in Y between that guy and that guy is this distance right here so let me draw a little triangle that distance right there is a change in Y or I could just transfer it to the y axis this is the change in Y that is your change in Y that distance so what is that distance it's F of B minus F of a so it equals F of B minus F of a F of B minus F of a that is your change in Y right here that's your change in Y now what is your change in X we have the slope is change in Y over change over change in X well what's our change in X what's this distance well we're taking this to be the first point so we took its Y minus the other points Y so to be consistent we're going to take this point X minus this points X so this point x-coordinate is B so it's going to be B minus a B minus a and just like that if you knew the equation of this line or if you had if you had the coordinates of these two points you would just plug them in right here and you would get your slope you would get your slope that's straight forward and that comes straight out of your algebra one class and let me just know just to make sure it's concrete for you if this was if this was the point let's say this was the point two three and let's say that this up here was the point five seven then if we wanted to find the slope of this line we would do seven minus three so seven minus three that would be our change in Y this would be seven and this would be three and then we would do that over five minus two five minus two because this would be a five and this would be a two and so this would be your change in X 5 minus two so 7 minus 3 is 4 and 5 minus 2 is 3 so your slope would be your slope would for over three now let's see if we can generalize this and this is what the the new concept that we're going to be learning as we delve into calculus let's see if we can generalize this somehow to a curve so let's say I have a curve we have to have a curve before we can generalize it to occur let me scroll down a little bit actually I want to leave this up here so that show you the similarity so let's say I have I'll keep it pretty general right now let's say I have a curve I'll make it a familiar-looking curve let's say it's the curve y is equal to x squared which looks something like that looks something like that and I want to find the slope let's say I want to find the slope at some point and actually before you've been talking about it let's even think about what it means to find the slope of a curve here the slope was the same the whole time right but on a curve your slope is changing and just to get an intuition for what that means is what's the slope over here your slope over here is the slope of the tangent line the line just barely touches it that's the slope over there it's a negative slope then over here your slope is still negative but it's a little bit less negative it goes like that I don't know if I did that drew that let me do it in a different color let me do it in purple so over here your slope is slightly next less negative it's a slightly less downward sloping line and then when you go over here at the zero point right here your slope is pretty much flat because it's the horizontal line y equals zero is tangent to this curve and then as you go to more positive X's then your slope starts increasing your slope starts increasing I'm trying to draw a tangent line and here it's increasing even more it's increased even more so your slope is changing the entire time and this is kind of the big change that happens when you go from a line to a curve a line your slope is the same the entire time you can take any two points in the line take the the change in Y over the change in X and you get the slope for the entire line but as you can see already it's going to be a little bit more nuanced when we do it for curve because it depends what point we're talking about we can't just say what is the slope for this curve the slope is different at every point along the curve it changes if we go up here just it's going to be even steeper it's going to look something like that something like that so let's try let's try a bit of an experiment and I know how this experiment turns out so it won't be too much of a risk let me draw it better than that that is my y-axis that is my y-axis and that's my x-axis X and let's call this we could call this Y or we call this the F of x-axis either way and let me draw my curve again so I'll just draw it in the positive coordinate like that that's my curve and what if I want to find the slope at what if I want to find the slope right there what can I do well based on our definition of a slope we need two points to find a slope right here I don't know how to find the slope with just one point so let's just say let's just call this one let's just call this point right here let's call that well that's going to be X we're going to be general this is going to be our point X but to find our slope according to our traditional algebra one definition slope we need two points so let's get another point in here let's get another point in here let's call let's just take a slightly very larger version of this X so let's say we want to take this let's say we hit let's say we have that point right there actually let me do it even further out just because it's going to get messy otherwise so let's say we have this point right here and the disk it's just H bigger than X or actually stead of saying H bigger let's just well let me just say H bigger so this is X plus h that's what that point is right there so what are they're going to be their corresponding Y coordinates on the curve well this is the curve of Y is equal to f of X so this point right here is going to be F of F of our particular X right here and maybe to show you that I'm taking a particular X maybe I'll do a little to zero here this is X not this is X not plus h this is f of X not and then what is what is this going to be up here this point up here that point up here its y-coordinate is going to be F of F of this x-coordinate which I shifted over a little bit is right there F of this x-coordinate which is f of X naught plus h that's its y-coordinate so what is the slope going to be between these two points that are relatively close to each other remember this isn't going to be the slope just at this point this is the slope of the line between these two points and if I were to actually draw it out it would actually be a secant line between two the curve so two intersect the curve twice once at this point and once at this point you can't see it if I drew if I blew it up a little bit it would look something like this it would look something like this it would look like that where this is our coordinate X naught f of X naught and up here is our coordinate is our coordinate for this point which would be the x coordinate would be X naught plus h and the y coordinate would be f of x naught plus h just whatever this function is we're evaluating it at this x-coordinate that's what it is so these are the two points so maybe a good start is just saying hey what is the slope of the secant line and just like we did in the previous example you find the change in X this is sorry the change in Y that'll be your change in Y and you divide that by your change in X you divide that by your change in X let me draw it here your change in Y would be that right here change in Y and then your change in X would be that right there so what is the slope going to be of the secant line the slope is going to be equal to let's start with this point up here just because it seems to be larger so it's we want to change in Y so this value right here this Y value is f of X naught plus h I just evaluated this guy up here it looks like a fancy term but all it means is look the slightly larger X evaluate its y-coordinate where it hit where the curve is at that at that value of x so that is going to be so the change in Y is going to be f of X naught plus h that's just the y-coordinate up here minus this y-coordinate over here so minus f of X naught and that's our change in so that equals our change in Y and you want to divide that by your change in X you want to divide that by your change in X so what is this this is the larger x value we started with this coordinates we start with its x-coordinate so it's x naught plus h x naught plus h minus this x-coordinate well we just picked a general number it's X naught X naught so that is the over your change in X just like that so this is the slope of the secant line we still haven't answered what the slope is right at that point or right at that point but maybe this will help us get there so if we simplify this so let me write it down like this the slope of the secant let me write that properly the slope of the secant line is equal to the value of the function at this point f of X naught plus h minus the value of the function here minus f of X naught so that just tells us the change in Y it's the exact same definition of slope we've always used over the change in X and we can simplify this we have x naught plus h minus X naught so X naught minus X naught cancel out so you have that over H so this is equal to our change in Y over change in X fair enough but I started it off saying I want to find the slope of the line at that point at this point right here this is the zoomed out version of it so what can I do well I defined this second point here is just the first point plus some H plus some H and we have something in our toolkit called a limit this H is just a general number right this H is just you know it could be ten it could be - it could be 0.02 it could be 1 times 10 to the negative 100 it can be an arbitrarily small number so what happens what would happen at least just theoretically if I were to take the limit as H approaches zero so if you know first maybe H is this fairly large number over here and then if I take H a little bit smaller than I'd be finding the slope of this secant line if I took H to be even a little bit smaller I'd be finding the slope of that secant line if H is a little smaller be finding the slope of that line so as H approaches zero I'll be getting closer and closer to finding the slope of the line right aft right at my point in question right at my point in question obviously if H is a large number my secant line is going to be way off from the slope at exactly that point right there but if H is 0.0001 if it's an infinitesimally small number then I'm going to get pretty close so what happens what happens if I take the limit as H approaches zero of this so the limit as H approaches zero of my secant slope of switch to green f of X naught plus h minus f of X naught that was my change in Y over my change in over H which is my change in X and now just to clarify something and sometimes you'll see it in different calculus books sometimes instead of an H they'll write a delta X here where this second point would have been defined as X naught plus Delta X and then this would have simplified to just Delta X over there and we'd be taking the limit as Delta X approaches zero the exact same thing H Delta X doesn't matter we're taking H as the difference between one x point and then the higher X point and the reason to take the limit as that approaches zero we could have called that Delta X just as easily but I am going to call this I'm going to call this thing which equals which we're saying it equals the slope of the tangent line and it does equal the slope of the tangent line I'm going to call this the derivative of F let me write that down the derivative derivative of F and I'm going to say that this is equal to F prime of X and this is going to be another function because remember the slope changes at every x value no matter what x value you pick the slope is going to be different it doesn't have to be but the way I drew this curve it is different it can be different so now you give me an x value in here I'll apply this formula over here and then I can tell you the slope at that point that all seems very using and maybe abstract at this point in the next video I'll actually do an example of calculating a slope and it'll make it everything a little bit more concrete
AP® is a registered trademark of the College Board, which has not reviewed this resource.