﻿>> Our discussion here is to develop a way to sort of generally generate slopes of tangent lines at particular points so let's generalize our thoughts here. Let's generalize the situation. Let's think that -- let me take out this secant line and we're going to just pick a general point x,y. And talk about how we can approach that point x,y like this. Right, here's the graph and let's just call this any old point x,y. I'm using this as a point but remember, this could be any point on the graph if I'm calling it x,y. All right, instead of now x,y, instead of using y, I'd like to use f of x. 'Cause f of x is y so I'll call this the point x, f of x and I'll pick another point and this other point I'm going to choose -- I want it to be very close here but just for -- to -- for notation purposes, I'll make it up here but it's arbitrarily close. I'm going to make it arbitrarily close to x by using this, I'm going to say this other point might be anywhere along the graph but I'm just going to call it x plus delta x. Delta x meaning just a little bitty distance; you see difference in x value are distance from x, I might say; x plus delta x. Now if that's the first component, if that's the first coordinate of that point, then f of that value, that is, f of x plus delta x would be the second component or the second coordinate. So we have the coordinates of a couple of points and now let's put this into the idea slope so the slope then would be -- let's see, difference in y components, remember from up here; difference in y components would be f of x plus delta x minus the other y component is f of x. And all of this is over -- let's see -- the first x value was x plus delta x and this is minus the other x component or first component is x. Now in the denominator, x minus x go out and we have simply delta x. Well, this kind of gives us an idea generally of what we want to do but you know, as delta x gets smaller and smaller and smaller, see we're talking about making littler and littler and littler, make that distance from x very, very tiny, then as we -- the closer we get to x, the better the approximation of the slope of the tangent line so we want the limit as -- in this situation, as delta x approaches zero, that's really what we're after to get that -- the slope of the tangent line to be exactly the amount that we want it to be. So let's go over here and talk about it in terms of limit; the slope then is -- the slope of that tangent line is the limit as delta x approaches zero of this fraction. It's the same fraction we had earlier and this will give us the slope of the tangent line and a point.