Local linearity and differentiability | Derivatives introduction | AP Calculus AB | Khan Academy

What we're going to do in this video is explore the relationship between local linearity at a point and differentiability at a point.

So, local linearity is this idea that if we zoom in sufficiently on a point, even a non-linear function that is differentiable at that point will actually look linear. So, let me show some examples of that.

Let's say we had y is equal to x squared. So, that's clearly a non-linear function, but we can zoom in on a point, and if we zoom sufficiently in, we will see that it looks roughly linear.

So, let's say we want to zoom in on the point (1, 1). So, zooming in on the point (1, 1), already it is looking roughly linear at that point. This property of local linearity is very helpful when trying to approximate a function around a point.

For example, we could take the derivative at the point (1, 1), use that as the slope of our tangent line, find the equation of the tangent line, and use that equation to approximate values of our function around x equals one. You might not need to do that for y is equal to x squared, but it could actually be very, very useful for a more complex function.

The big takeaway here is that at the point (1, 1), it is displaying this idea of local linearity, and it is also differentiable at that point.

Now, let's look at another example of a point on a function where we aren't differentiable, and we also don't see the local linearity. For example, let's do the absolute value of x, and let me shift it over a little bit just so that we don't overlap as much.

All right, so the absolute value of x minus 1 actually is differentiable as long as we're not at this corner right over here. As long as we're not at the point (1, 0), for any other x value, it is differentiable. But right at x equals 1, we've talked in other videos how we aren't differentiable there.

We can use this local linearity idea to test it as well. Once again, this is not rigorous mathematics, but it is to give you an intuition. No matter how far we zoom in, we still see this sharp corner.

It would be hard to construct a unique tangent line that goes through the point (1, 0). I can construct an actual infinite number of lines that go through (1, 0) but that do not go through the rest of the curve.

So, notice wherever you see a hard corner, like we're seeing at (1, 0) in this absolute value function, that's a pretty good indication that we are not going to be differentiable at that point.

Now, let's zoom out a little bit and let's take another function. Let's take a function where the differentiability or the lack of differentiability is not because of a corner but it's because as we zoom in, it starts to look linear, but it starts to look like a vertical line.

A good example of that would be the square root of (4 - x²). So, that's the top half of a circle of radius 2. Let's focus on the point (2, 0) because right over there, we actually are not differentiable.

If we zoom in far enough, we see right at (2, 0) that we are approaching what looks like a vertical line. So, once again, we would not be differentiable at (2, 0).

Now, another thing I want to point out: all of these you really didn't have to zoom in too much to appreciate that, hey, I got a corner here on this absolute value function, or at (2, 0) or at (-2, 0). Something a little bit stranger than normal is happening there, so maybe I'm not differentiable.

But there are some functions that we don't see as typically in an algebra, precalculus, or calculus class, but it can look like a hard corner from a zoomed-out perspective. But as we zoom in once again, we'll see the local linearity, and they are also differentiable at those points.

So, a good example of that—let me actually get rid of some of these just so we can really zoom in—let's say y is equal to x to the tenth power. It's starting to look a little bit like a corner there.

Let's make it to the hundredth power. Well, now it's looking even more like a corner there. Let me go to the thousandth power just for good measure. So, at this scale, it looks like we have a corner at the point (1, 0).

Now, this curve actually does not go to the point (1, 0). If x is 1, then y is going to be 1. As we zoom in, what looks like a hard corner is going to soften.

That's good because this function is actually differentiable at every value of x. It's a little bit more exotic than what we typically see, but as we zoom in, we'll actually see that.

Let's just zoom in on what looks like a fairly hard corner. But if we zoom sufficiently enough— even at the part that looks like the hardest part of the corner, the real corner—we'll see that it starts to soften and it curves.

If we zoom in sufficiently, it will actually look like a line. It's hard to believe when you're really zoomed out and I'm going at the point that really looked like a corner from a distance.

But as we zoom in, once again, this local linearity emerges, and it's a non-vertical line. So, once again, this is true at any point on this curve that we are going to be differentiable.

The whole point here is sometimes you might have to zoom in a lot. A tool like Desmos, which I'm using right now, is very helpful for doing that. This isn't rigorous mathematics, but it gives you an intuitive sense that if you zoom in sufficiently and you start to see a curve looking more and more like a line, it's a good indication that you are differentiable.

If you keep zooming in and it still looks like a hard corner, or if you zoom in and it looks like the tangent might be vertical, well then some questions should arise in your brain.