Formal definition of the derivative as a limit
Worked example: Derivative from limit expression
The alternate form of the derivative of the function f, at a number a, denoted by f prime of a, is given by this stuff. Now this might look a little strange to you, but if you really think about what it's saying, it's really just taking the slope of the tangent line between a comma f of a. So let's imagine some arbitrary function like this. Let's say that that is-- well I'll just write that's our function f. And so you could have the point when x is equal to a-- this is our x-axis-- when x is equal to a, this is the point a, f of a. You notice a, f of a. And then we could take the slope between that and some arbitrary point, let's call that x. So this is the point x, f of x. And notice, the numerator right here, this is just our change in the value of our function. Or you could view that as the change in the vertical axis. So that would give you this distance right over here. That's what we're doing up here in the numerator. And then in the denominator, we're finding the change in our horizontal values, horizontal coordinates. Let me do that in a different color. So the change in the horizontal, that's this right over here. And then they're trying to find the limit as x approaches a. So as x gets closer and closer and closer and closer to a, what's going to happen is, is that when x is out here, we have this secant line. We're finding the slope of this secant line. But as x gets closer and closer, the secant lines better and better and better approximate the slope of the tangent line. Where the limit, as x approaches a, but doesn't quite equal a, is going to be-- this is actually our definition of our derivative. Or I guess the alternate form of the derivative definition. And this would be the slope of the tangent line, if it exists. So with that all that out the way, let's try to answer their question. With the Alternative Form of the Derivative as an aid, make sense of the following limit expression by identifying the function f and the number a. So right here, they want to find the slope of the tangent line at 5. Here they wanted to find the slope of the tangent line at a. So it's pretty clear that a is equal to 5. And that f of a is equal to 125. Now what about f of x? Well here, it's a limit of f of x minus f of a. Well here it's the limit as x to the third minus 125. And this makes sense. If f of x is equal to x to the third, then it makes sense that f of 5 is going to be 5 to the third, is going to be 125. And we're also taking up here the limit as x approaches a. Here we're taking the limit as x approaches 5. So this is the derivative of the function f of x is equal to x to the third. Let me write that down in the green color. x to the third at the number a is equal to 5. And so we can imagine this. Let's try to actually graph it, just so that we can imagine it. Actually, I'll do it out here, where I have a little bit better contrast with the colors. So let's say that is my y-axis. Let's say that this is my x-axis. I'm not going to quite draw it to scale. Let's say this right over here is the 125. Or y, this is when y equals 125. This is when x is equal to 5, so they're clearly not at the same scale. But the function is going to look something like this. We know what x to the third looks like, it looks something like this. So here, our a is equal to 5. This point right over here is 5, 125. And then we're taking the slope between that point and an arbitrary x-value. Or I should say an arbitrary other point on the curve. So this right over here would be the point, we could call that x, x to the third. We know that f of x is equal to x to the third. And let me make it clear. This is a graph of y is equal to x to the third. And so this expression, right over here, all of this, this is the slope between these two points. And as we take the limit as x approaches 5, so right now this is our x, as x gets closer and closer and closer to 5, the secant lines are going to better and better approximate the slope of the tangent line at x equals 5. So the slope of a tangent line would look something like that.