AP®︎/College Calculus AB
- Formal definition of the derivative as a limit
- Formal and alternate form of the derivative
- Worked example: Derivative as a limit
- Worked example: Derivative from limit expression
- Derivative as a limit
- The derivative of x² at x=3 using the formal definition
- The derivative of x² at any point using the formal definition
- Finding tangent line equations using the formal definition of a limit
Sal finds the limit expression for the derivative of f(x)=x² at any point x and simplifies it into a simpler expression (spoiler: it's 2x). Created by Sal Khan.
In the last video, we found the slope at a particular point of the curve y is equal to x squared. But let's see if we can generalize this and come up with a formula that finds us the slope at any point of the curve y is equal to x squared. So let me redraw my function here. It never hurts to have a nice drawing. So that is my y-axis. That is my x-axis right there. My x-axis. Let me draw my curve. It looks something like that. You've seen that multiple times. This is y is equal to x squared. So let's be very general right now. Remember, if we want to find-- let me just write the definition of our derivative. So if we have some point right here-- let's call that x. So we want to be very general. We want to find the slope at the point x. We want to find a function where you give me an x and I'll tell you the slope at that point. We're going to call that f prime of x. That's going to be the derivative of f of x. But all it does is, look, f of x, you give-- it's a function that you give it an x, and it tells you the value of that. And we draw the curve here. With f of x, you give that same x but it's not going to tell you the value of the curve. It's not going to say, oh, this is your f of x. It's going to give you the value of the slope of the curve at that point. So f of x, if you put it into that function, it should tell you, oh, the slope at that point, is equal to-- you know, if you put 3 there, you'll say, oh, the slope there is equal to 6. We saw that in the last example. So that's what we want to do. And we saw on the last-- I think it was 2-- videos ago, that we defined f prime of x to be equal to-- just the-- well, I'll write it this way. It's the slope of the secant line between x and some point that's a little bit further away from x. So the slope of the secant line is change in y. So it's the y value of the point that's a little bit further away from x. So f of x plus h minus the y value at x, right? Because this is right here. This is f of x. So minus f of x. All of that over the change in x. So if this is x plus h here, the change in x is x plus h minus x. Or this distance right here is just h. The change in x is going to be equal to h. So that's just slope of the secant line, between any 2 points like that. And we said, hey, we could find the slope of the tangent line if we just take the limit of this as it approaches-- as h approaches 0. And then we'll be finding the slope of the tangent line. Now let's apply this idea to a particular function, f of x is equal to x squared. Or y is equal to x squared. So here, we could have the point-- we could consider this to be the point x-- x squared. So f of x is just equal to x squared. And then this would be the point-- let me do it in a more vibrant color. This is the point x plus h-- that's this point right here. It's a little bit further down. And then x plus h squared. And you know, in the last video, we did this for a particular x. We did it for 3. But now I want a general formula. You give me any x and I won't have to do what I did in the last video for any particular number. I'll have a general function. You give me 7, I'll tell you what the slope is at 7. You give me negative 3, I'll tell you what the slope is at negative 3. You give me 100,000, I'll tell you what the slope is at 100,000. So let's apply it here. So we want to find the change in y over the change in x. So first of all, the change in y is this guy's y value, which is x plus h squared. That's this guy's y value right here. That's this right here. That's x plus h squared. I just took x plus h, evaluated, I squared it, and that's its point on the curve. So it's x plus h squared. So that's there right there. And then what's this value? f of x right here is equal to-- I know it's getting messy-- is equal to x squared. If you take your x, you evaluate the function at that point, you're going to get x squared. So it's equal to minus x squared. This is your change in y. That's this distance right there. And just to relate it to our definition of a derivative, this blue thing right here is equivalent to this thing right here. We just evaluated our function. Our function is f of x is equal to x squared. We just evaluated when x is equal to x plus h. So if you have to square it, if I put an a there, it'd be a squared. If I put an apple there, it'd be apple squared. If I put an x plus h in there, it's going to be x plus h squared. So this is that thing. And then, this thing right here is just the function evaluated at the point in question. Right there. So this is our change in y. And let's divide that by our change in x. Our change in x-- if this is x plus h and this is just x, our change in x is just going to be h. So that's where we get that term from. So this is just a slope between these 2 points. This is just a slope between those two points. But, of course, we want to find-- the limit at this point gets closer and closer to this point, and this point gets closer and closer to that point. So this becomes a tangent line. So we're going to take the limit as h approaches 0, and this is our f prime of x. And this is the exact same definition of this, instead of being general and saying, for any function, we know what the function was. It was f of x is equal to x squared. So we actually applied it. Instead of f of x, we wrote x squared. Instead of f of x plus h, we wrote x plus h squared. So let's see if we can evaluate this limit. So this is going to be equal to the limit as h approaches 0 to square this out. I'll do it in the same color. That's x squared plus 2xh plus h squared, and then we have this minus x squared over here. I just multiplied this guy out over here. And then all of that is divided by h. Now let's see if we can simplify this a little bit. Well, you immediately see you have an x squared and you have a minus x squared, so those cancel out. And then we can divide the numerator and the denominator by h. So this simplifies to-- so we get f prime of x is equal to-- if we divide the numerator and the denominator by h-- we get 2x plus h. I'm sorry, I forgot my limit. It equals the limit. Very important. Limit as h approaches 0 of divide everything by h, and you get 2x plus h squared divided by h is h. And if you remember the last video, when we did it with a particular x, we said x is equal to 3, we got 6 plus delta x here. Or 6 plus h here, so it's very similar. So if you take the limited h approaches 0 here, that's just going to disappear. So this is just going to be equal to 2x. So we just figured out that if f of x-- this is a big result. This is exciting! That if f of x is equal to x squared, f prime of x is equal to 2x. That's what we just figured out. And I wanted to make sure you understand how to interpret this. f of x, if you give me a value, is going to tell you the value of the function at that point. At prime of x it's going to tell you the slope at that point. Let me draw that. Because this is a key realization. And you might, you know, it's kind of maybe initially unintuitive to think of a function that gives us the slope, at any point, of another function. So it looks like this. Let me draw a little neater than that. Ah, it's still not that neat. That's satisfactory. Let me just draw it in the positive coordinate. Well, I'll just draw the whole-- the curve looks something like that. Now this is the curve of f of x. This is the curve of f of x is equal to x squared. Just like that. So if you give me a point. You give me the point 7. You apply, you put it in here, you square it. And it is mapped to the number 49. So you get the number 49 right there. This is the number 7, 49. You're used to dealing with functions right there. But what is f prime of 7? f prime of 7. You say, 2 times 7 is equal to 14. What is this 14 number here? What is this thing? Well, this is the slope of the tangent line at x is equal to 7. So if I were to take that point and draw a tangent line-- a point that just grazes our curve-- if I were to just draw a tangent line. That wasn't tangent enough for me. So that's my tangent line right there. You get the idea. The slope of this guy-- you do your change in y over your change in x-- is going to be equal to 14. The slope of the curve at y is equal to 7-- is a pretty steep curve. If you wanted to find the slope, let's say that this is y-- let's say it's x is equal to 2. I said at x is equal to 7, the slope is 14. At x is equal to 2, what is the slope? Well, you figure out f prime of 2, which is equal to 2 times 2, which is equal to 4. So the slope here is 4. You could say m is equal to 4. m for slope. What is f prime of 0? f prime. We know that f of 0 is 0, right? 0 squared is 0. But what is f prime of 0? Well, 2 times 0 is 0. That's also equal to 0. But what does that mean? What's the interpretation? It means the slope of the tangent line is 0. So a 0 sloped line looks like this. Looks just like a horizontal line. And that looks about right. A horizontal line would be tangent to the curve at y equals 0. Let's try another one. Let's try the point minus 1. So let's say we're right there. x is equal to minus 1. So f of minus 1, you just square it. Because we're dealing with x squared. So it's equal to 1. That's that point right there. What is f prime of minus 1? f prime of minus 1 is 2 times minus 1. 2 times minus is minus 2. What does that mean? It means that the slope of the tangent line at x is equal to 1, to this curve, to the function, is minus 2. So if I were to draw the tangent line here-- the tangent line looks like that-- and look, it is a downward sloping line. And it makes sense. The slope here is equal to minus 2.