Derivative as slope of curve | Derivatives introduction | AP Calculus AB | Khan Academy

What I want to do in this video is a few examples that test our intuition of the derivative as a rate of change or the steepness of a curve, or the slope of a curve, or the slope of a tangent line of a curve, depending on how you actually want to think about it.

So here it says F Prime of five. So this notation Prime, this is another way of saying, well, what's the derivative? Let's estimate the derivative of our function at five.

And when we say F prime of five, this is the slope, slope of the tangent line at five, or you could view it as the rate of change of Y with respect to X, which is really how we define slope with respect to X of our function f.

So let's think about that a little bit. We see they put the point, the point five comma F of 5 right over here.

And so if we want to estimate the slope of the tangent line, or if we want to estimate the steepness of this curve, we could try to draw a line that is tangent right at that point.

And so let me see if I can do that. So if I were to draw a line starting there, and if I just wanted to make it tangent, it looks like it would do something like that. That right at that point, that looks to be about how steep that curve is.

Now what makes this an interesting thing and nonlinear is that it's constantly changing the steepness. It's very low here, and it gets steeper and steeper and steeper as we move to the right for larger and larger X values.

But if we look at the point in question when X is equal to 5, remember F prime of five would be, if we were estimating it, this would be the slope of this line here.

And the slope of this line, it looks like for every time we move one in the X direction, we're moving two in the Y direction. Delta Y is equal to 2 when Delta X is equal to one.

So our change in Y with respect to X, at least for this tangent line here, which would represent our change in Y with respect to X right at that point, is going to be equal to 2 over 1, or two.

And they told us to estimate it, but all of these are way off. Having a negative two derivative would mean that as we increase our X, our Y is decreasing.

So if our curve looks something like this, we would have a slope of -2. Having slopes in this range, a positive of 0.1, that would be very flat.

Some down here we might have a slope closer to 0.1 negative 0.1. That might be closer on this side where it's downward sloping but very close to flat.

A slope of zero would be right over here at the bottom where, right at that moment, as we change X, Y is not increasing or decreasing. The slope of the tangent line right at that bottom point would have a slope of zero.

So I feel really good about that response. Let's do one more of these.

So all right, so they're telling us to compare the derivative of g at 4 to the derivative of g at 6 and which of these is greater.

And like always, pause the video and see if you could figure this out.

Well, this is just an exercise. Let's see if we were to make a line that indicates the slope there, and you can view this as a tangent line.

So let me try to do that. So that wouldn't, that doesn't do a good job. So right over here at... so that looks like a pretty, I think I could do a better job than that.

No, that's too shallow. Let's see. Not shallow, though. That's too flat. So let me try to really... okay, that looks pretty good.

So that line that I just drew seems to be indicative of the rate of change of Y with respect to X, or the slope of that curve, or that line. You could view it as a tangent line so that we can think about what its slope is going to be.

And then if we go further down over here, this one is, it looks like it is steeper, but in the negative direction.

So it looks like it is steeper for sure, but it's in the negative direction. As we increase, think of it this way, as we increase X here, it looks like we are decreasing Y by about one.

So it looks like G prime of 4, G prime of 4, the derivative when X is equal to 4 is approximately, I'm estimating it, -1, while the derivative here, when we increase X, if we increase X by 1, it looks like we're decreasing Y by close to three.

So G prime of 6 looks like it's closer to -3. So which one of these is larger? Well, this one is less negative, so it's going to be greater than the other one.

And you could have done this intuitively. If you just look at the curve, this is some type of a sinusoid here. You have right over here, the curve is flat.

It's, you have right at that moment, you have no change in Y with respect to X. Then it starts to decrease at a f, and then it decreases at an even faster rate.

Then it decreases at a faster rate, then it starts, it's still decreasing, but it's decreasing at slower and slower rates. Decreasing at slower rates, and right at that moment, it's not.

You have your slope of your tangent line is zero, then it starts to increase, increase, so on and so forth, and it just keeps happening over and over again.

So you could also think about this in a more intuitive way.