Formal definition of derivative
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Worked example: Derivative as a limit
Let's say that f of x is equal to the natural log of x, and we want to figure out what the slope of the tangent line to the curve f is when x is equal to the number e. So here, x is equal to the number e. The point e comma 1 is on the curve. f of e is 1. The natural log of e is 1. And I've drawn the slope of the tangent line, or I've drawn the tangent line. And we need to figure out what the slope of it is, or at least come up with an expression for it. And I'm going to come up with an expression using both the formal definition and the alternate definition. That will allow us to compare them a little bit. So let's think about first the formal definition. So the formal definition wants us to find an expression for the derivative of our function at any x. So let's say that this is some arbitrary x right over here. This would be the point x comma f of x. And let's say that this is-- let's call this x plus h. So this distance right over here is going to be h. This right over here is going to be the point, x plus h f of x plus h. Now, the whole underlying idea of the formal definition of limits is to find the slope of the secant line between these two points, and then take the limit as h approaches 0. As h gets closer and closer, this blue point is going to get closer and closer and closer to x. And this point is going to approach it on the curve. And the secant line is going to become a better and better and better approximation of the tangent line at x. So let's actually do that. So what's the slope of the secant line? Well, it's the change in your vertical axis, which is going to be f of x plus h minus f of x-- over the change in your horizontal axis. And that's x plus h minus x. And we see here the difference is just h. Over h. And we're going to take the limit of that as h approaches 0. So in the case when f of x is the natural log of x, this will reduce to the limit as h approaches 0. f of x plus h is the natural log of x plus h minus the natural log of x, all of that over h. So this right over here, for our particular f of x, this is equal to f prime of x. So if we wanted to evaluate this when x is equal to e, then everywhere we see an x we just have to replace it with an e. This is essentially expressing our derivative as a function of x. It's kind of a crazy-looking function of x. You have a limit here and all of that. But every place you see an x, like any function definition, you can replace it now with an e. So we can-- let me just do that. Whoops. I lost my screen. Here we go. So we could write f prime of e is equal to the limit as h approaches 0 of natural log-- let me do it in the same color so we can keep track of things-- natural log of e plus h-- I'll just leave that blank for now-- minus the natural log of e, all of that over h. So just like that. This right over here, if we evaluate this limit-- if we're able to and we actually can-- if we are able to evaluate this limit, this would give us the slope of the tangent line when x equals e. This is doing the formal definition. Now let's do the alternate definition. The alternate definition-- if you don't want to find a general derivative expressed as a function of x like this and you just want to find the slope at a particular point, the alternate definition kind of just gets straight to the point there. So what they say is hey, look, let's imagine some other x value here. So let's imagine some other x value. This right over here is the point x comma-- well, we could say f of x or we could even say the natural log of x. What is the slope of the secant line between those two points? Well, it's going to be your change in y values. So it's going to be natural log of x minus 1-- let me do that red color-- over your change in x values. That's x minus e. So that's the slope of the secant line between those two points. Well, what if you want to get the tangent line? Well, let's just take the limit as x approaches e. As x gets closer and closer and closer, these points are going to get closer and closer and closer, and the secant line is going to better approximate the tangent line. So we're just going to take the limit as x approaches e. So either one of this. This is using the formal definition of a limit. Let me make it clear that that h does not belong part of it. So we could either do it using the formal definition or the alternate definition of the derivative.