Differentiability at a point: graphical | Derivatives introduction | AP Calculus AB | Khan Academy

The graph of function f is given below. It has a vertical tangent at the point (3, 0). So (3, 0) has a vertical tangent. Let me draw that. So it has a vertical tangent right over there and a horizontal tangent at the point (0, -3). (0, -3) has a horizontal tangent right over there and also has a horizontal tangent at (6, 3). So (6, 3), let me draw the horizontal tangent just like that.

Select all the x values for which f is not differentiable. Select all that apply. So f prime... I’ll write it in shorthand, so we say no f prime. Under it, it's going to happen under three conditions. The first condition you could say, well, we have a vertical tangent. Vertical tangent... why is a vertical tangent a place where it's hard to define our derivative? Well, remember, our derivative is we're really trying to find a rate of change of y with respect to x. But when you have a vertical tangent, you change your x a very small amount; you have an infinite change in y, either in the positive or the negative direction.

So that's one situation where you have no derivative. They tell us where we have a vertical tangent here, where x is equal to 3. So we have no... f is not differentiable at x equals 3 because of the vertical tangent. You might say, what about horizontal tangents? No, horizontal tangents are completely fine. Horizontal tangents are places where the derivative is equal to 0. So f prime of 6 is equal to 0; f prime of 0 is equal to 0.

What are other scenarios? Well, another scenario where you're not going to have a defined derivative is where the graph is not continuous. Not continuous. We see right over here at x equals -3, our graph is not continuous. So x equals -3; it's not continuous. And those are the only places where f is not differentiable that they’re giving us options on. We don't know what the graph is doing to the left or the right. These are, I guess, interesting cases, but they haven't given us those choices here.

And we already said at x equals 0, the derivative is 0; it's defined, it's differentiable there. And at x equals 6, the derivative is 0. We have a flat tangent, so once again, it's defined there as well.

Let's do another one of these. Oh, and actually I didn't include... I think that this takes care of this problem. But there's a third scenario in which we have, I'll call it a sharp turn. A sharp turn... this isn't the most mathy definition right over here, but it's easy to recognize. A sharp turn is something like that or like... well, that doesn't look too sharp, or like this. The reason why I think where you have these sharp bends or sharp turns, as opposed to something that looks more smooth like that, the reason why we're not differentiable there is, as we approach this point, from either side, we have different slopes.

Notice our slope is positive right over here. Whereas x increases, y is increasing. Well, our slope is negative here. So as you try to find the limit of our slope as we approach this point, it's not going to exist because it's different on the left-hand side and the right-hand side. So that's why the sharp turns... I don't see any sharp turns here, so it doesn't apply to this example.

Let's do one more example. And actually, this one does have some sharp turns, so this could be interesting. The graph of function f is given to the left right here. It has a vertical asymptote at x equals -3; we see that. And horizontal asymptotes at y equals 0. Yep! This end of the curve, as x approaches negative infinity, it looks like y is approaching 0, and has another horizontal asymptote at y equals 4. As x approaches infinity, it looks like our graph is trending down to y equals 4.

Select the x values for which f is not differentiable. So first of all, we could think about vertical tangents. Doesn't seem to have any vertical tangents. Then we could think about where we are not continuous. Well, we're definitely not continuous where we have this vertical asymptote right over here, so we're not continuous at x equals -3. We're also not continuous at x equals 1.

And then the last situation where we are not going to be differentiable is where we have a sharp turn, or you could kind of view it as a sharp point on our graph. I see a sharp point right over there. Notice as we approach from the left-hand side, the slope looks like... it looks like a constant. I don't know, is this like a positive 3/2? Well, right as we go to the right side of that, it looks like our slope turns negative.

So if you were to try to find the limit of the slope as we approach from either side, which is essentially what you're trying to do when you try to find the derivative—well, it's not going to be defined because it's different on either side. So we also... the f is also not differentiable at this x value that gives us that little sharp point right over there.

And if you were to graph the derivative, which we will do in future videos, you will see that the derivative is not continuous at that point. So let me mark that off. And then we could check x equals 0. x equals 0 is completely cool. We're at a point that our tangent line is definitely not vertical. We're definitely continuous there. We definitely do not have a sharp point or edge, so we're completely cool at x equals 0.