Derivative as a concept | Derivatives introduction | AP Calculus AB | Khan Academy

You are likely already familiar with the idea of a slope of a line. If you're not, I encourage you to review it on Khan Academy.

But all it is, it's describing the rate of change of a vertical variable with respect to a horizontal variable. So, for example, here I have our classic y-axis in the vertical direction and x-axis in the horizontal direction.

And if I wanted to figure out the slope of this line, I could pick two points—say that point and that point. I could say, okay, from this point to this point, what is my change in X?

Well, my change in X would be this distance right over here, change in X—the Greek letter Delta, this triangle here—it's just shorthand for change. So, change in X. And I could also calculate the change in y. So, this point going up to that point, our change in y would be this right over here, our change in y.

And then we would define slope—or we have defined slope—as change in y over change in X. So, slope is equal to the rate of change of our vertical variable over the rate of change of our horizontal variable. It's sometimes described as rise over run.

For any line, it's associated with a slope because it has a constant rate of change. If you took any two points on this line, no matter how far apart or no matter how close together, anywhere they sit on the line, if you were to do this calculation, you would get the same slope. That's what makes it a line.

But what's fascinating about calculus is we're going to build the tools so that we can think about the rate of change not just of a line—which we've called slope in the past; we can think about the rate of change, the instantaneous rate of change of a curve—of something whose rate of change is possibly constantly changing.

So, for example, here's a curve where the rate of change of y with respect to x is constantly changing. Even if we wanted to use our traditional tools, if we said, okay, we can calculate the average rate of change, let's say between this point and this point, well, what would it be?

Well, the average rate of change between this point and this point would be the slope of the line that connects them. So, it'd be the slope of this line, the secant line. But if we pick two different points, if we pick this point and this point, the average rate of change between those points all of a sudden looks quite different; it looks like it has a higher slope.

So, even when we take the slopes between two points on the line, the secant lines, you can see that those slopes are changing. But what if we wanted to ask ourselves an even more interesting question: What is the instantaneous rate of change at a point?

So, for example, how fast is y changing with respect to x exactly at that point, exactly when x is equal to that value? Let's call it x1. Well, one way you could think about it is what if we could draw a tangent line to this point—a line that just touches the graph right over there—and we can calculate the slope of that line.

Well, that should be the rate of change at that point, the instantaneous rate of change. So, in this case, the tangent line might look something like that. If we know the slope of this, well, then we could say that that's the instantaneous rate of change at that point.

Why do I say instantaneous rate of change? Well, think about the video on the sprinter's, the Usain Bolt example. If we wanted to figure out the speed of Usain Bolt at a given instant, well maybe this describes his position with respect to time.

If y was position and x is time, usually you would see t as time, but let's say x is time. So then if we're talking about right at this time, we're talking about the instant instantaneous rate. And this idea is the central idea of differential calculus, and it's known as a derivative—the slope of the tangent line, which you could also view as the instantaneous rate of change. I'm putting an exclamation mark because it's so conceptually important here.

So how can we denote a derivative? One way is known as Leibniz's notation, and Leibniz is one of the fathers of calculus along with Isaac Newton. In his notation, you would denote the slope of the tangent line as equaling dy over dx.

Now, why do I like this notation? Because it really comes from this idea of a slope, which is change in y over change in x. As you'll see in future videos, one way to think about the slope of the tangent line is, well, let's calculate the slope of secant lines—let's say between that point and that point—but then let's get even closer—say that point and that point—and then let's get even closer, and that point and that point—and then let's get even closer and see what happens as the change in x approaches zero.

And so using these d's instead of deltas, this was Leibniz's way of saying, "Hey, what happens if my changes in, say, x become close to zero?" So this idea, this is known as sometimes differential notation. Leibniz's notation is instead of just change in y over change in x, super small changes in y for a super small change in x, especially as the change in x approaches zero.

And as you will see, that is how we will calculate the derivative. Now, there are other notations. If this curve is described as y is equal to f of x, the slope of the tangent line at that point could be denoted as equaling f prime of x1.

So this notation takes a little bit of time getting used to. The "f prime" notation is saying f prime represents the derivative. It's telling us the slope of the tangent line for a given point. So if you input an x into this function, into f, you’re getting the corresponding y value. If you input an x into f prime, you're getting the slope of the tangent line at that point.

Now, another notation that you'll see—less likely in a calculus class but you might see in a physics class—is the notation y with a dot over it. So you could write this as y with a dot over it, which also denotes the derivative. You might also see y prime; this would be more common in a math class.

Now, as we march forward in our calculus adventure, we will build the tools to actually calculate these things. And if you're already familiar with limits, they will be very useful, as you can imagine, because we're really going to be taking the limit of our change in y over change in x as our change in x approaches zero.

And we're not just going to be able to figure it out for a point; we're going to be able to figure out general equations that describe the derivative for any given point. So, be very, very excited!