Connecting f, f', and f'' graphically (another example) | AP Calculus AB | Khan Academy

We have the graph of three functions here, and we're told that one of them is the function ( f ), one is its first derivative, and then one of them is the second derivative. We just don't know which one is which. So, like always, pause this video and see if you can figure it out.

All right, now the way I'm going to tackle it is I'm going to look at each of these graphs and try to think what would their derivatives look like. So for this first one, we can see our derivative right over here. Our slope of our tangent line would be a little bit negative, and then it gets more and more and more negative. As we approach this vertical asymptote right over here, it looks like it's approaching negative infinity. So the derivative would actually—over here it would be a little bit less than zero—but then it would get more and more and more negative, and then it would approach negative infinity.

So, it would have a similar shape, general shape, to the graph itself, at least to the left of this vertical asymptote. Now, what about to the right of the vertical asymptote? Right to the right of the vertical asymptote, it looks like the slope of the tangent line is very negative. It's very negative, but then it becomes less and less and less negative, and it looks like it is approaching zero. So on this side, the derivative starts out super negative, and then it looks like the derivative is going to asymptote towards zero, something like that.

Based on what we just sketched, it looks like this right graph is a good candidate for the derivative of this left graph. You might say what's wrong with this blue graph? Well, this blue graph out here, notice it's positive. So if this were the derivative of the left graph, that means that the left graph would need a positive slope out here, but it doesn't have a positive slope. It's a very—it's a slightly negative slope becoming super negative. And so right here, we're slightly negative, and then we become very negative.

So maybe this is—let's call this ( f ), and maybe this is ( f' ). This is ( f' ) right over here. Now let's look at this middle graph. What would its derivative do? So over here, our slope is slightly negative, and then it becomes more and more and more and more negative. So the derivative of this might look like—it has to be slightly negative, but then it gets more and more and more and more negative as we approach that vertical asymptote.

On the right side of the vertical asymptote, our derivative is very positive here, and then it gets less and less and less and less positive. So we start—our derivative would be very positive, and then it would get less and less and less and less positive. It looks like it might—the slope here might be asymptoting towards zero, so our graph might look something like that.

Well, the left graph right here looks a lot like what I just sketched out as a candidate derivative for this blue graph, for this middle graph. So I would say that this is ( f ), then this is the derivative of that, which would make it ( f' ), and then we already established that this right graph is the derivative of the left one. So if it's the derivative of ( f' ), it's not ( f' ) itself; it's the second derivative.

So I feel pretty good about that. Just for good measure, we could think about what the derivative of this graph would look like. Here, the slope is slightly negative, but then it gets more and more and more and more and more negative. So the derivative would have a similar shape here. Then here, our derivative would be very positive, and it gets less and less and less and less positive. So we start very positive, and then it gets less and less and less and less positive.

As a general shape, it actually does look a lot like this first graph. But the reason why I'm not going to say that this first graph is the derivative of the right-hand graph is because this right-hand graph was the only good candidate that we had for the derivative of the left-hand graph. So I feel pretty good with what we selected, that this middle one is ( f ), the left one is the first derivative, and the right one is the second derivative.