Derivative as a function
Connecting f and f' graphically
So what we have plotted here is some function-- let's call it f of x-- and its derivative, f prime of x. And what we need to figure out is which one is f of x and which one is f prime of x? So let's take a stab at it. Let's think about what would be the situation if the green function were f of x. So let's see if this works out. If the green function were f of x, does the orange function here, or the yellow function, could that be f prime of x? So let's think about what's happening to this green function at different points. So this green function right over here, right at this point if we start at the left, has a positive slope. If this orange function were f prime of x, if it were the derivative of the green function, then it would have to be positive because the green function's slope is positive at that point. But we see that it's not positive. So it's pretty clear that the green function cannot be f of x, and the yellow function cannot be its derivative, because if this was its derivative, it would be positive here. So that quickly, we found out that that can't be the case. But let's see if it could work out the other way. So it's starting to feel-- just ruling that situation out-- that maybe that this is f of x and the green function is f prime of x. So let's see if this holds up to scrutiny. So what we have when we start off at the left, f of x, or what we think is f of x, has a reasonably positive slope. Is that consistent? Well, yeah, sure. Our green function is positive. In fact, at the point, it's telling us that the slope of the tangent line is around 2 and 1/2. And it actually does look like the slope of the tangent line is exactly 2 and 1/2 of this function right here. Actually, let me erase this, just so we don't look like we're trying to take the slope of the tangent line of the derivative. So it looks just like that. So we see the slope of the tangent line right over here looks like about 2 and 1/2, and the value of this function up here looks like it's about 2 and a 1/2. So, so far this green function looks like a pretty good candidate for the derivative of this yellow function. But let's keep going here. So let's think about what happens as we move to the right. So here, let's see. It looks like the slope of this yellow function-- let me just use a color we can see-- it keeps going up. It keeps going up, keeps going up. And then at some point, it reaches some maximum slope, and then it starts to go down again. The slope starts to go down again, all the way to the slope going all the way down to 0 right over here. Well, does this green function describe that? Well, let's see. The slope is positive and increasing up to this point, which seemed pretty consistent with what we just experienced. Then the slope stays positive, but it's positive and decreasing. And that's what we saw here. The slope is positive and decreasing all the way to the slope getting to 0 at this maximum point here. And we see, indeed, that on this green function, the green function hits 0. So it seems like it's doing a pretty good job of plotting the slope of the tangent line of the orange function. And then our slope becomes more and more negative. And then it hits some point, some minimum point right over here. The slope hits some minimum point right over here. And then it becomes less and less negative. Let's see how well I can draw it. The slope becomes less and less negative until it hits a 0 slope again. And then it starts becoming positive until it hits some maximum slope. But then it stays positive, but it becomes less positive. It becomes less and less positive. So it looks pretty clear that the orange function is f of x, and the green function is f prime of x.