 So, a useful thing we want to be able to do is to find the derivative from the graph of a function. And so, the situation arises as follows. Suppose we have the graph of y equals f of x. We might want to be able to sketch the graph of y equals f prime of x without having to differentiate f of x. And the reason that we might need to do this is we might only have the graph. We might not actually have a formula that we can apply a derivative rule to. So, how can we do that? Well, we could start with a perfectly accurate graph of the derivative, but that's kind of difficult. So, we'll start with the stick figure. Now, well, let's start with a rough sketch of the graph of y equals f prime of x. And so, we'll start with a rough sketch. And then later on, we'll refine our graph and make it somewhat more artistic. The key thing here is that we have to start with that rough sketch before we can actually make the refined sketch of our graph. So, let's take a look at some problems here. So, here's the graph of y equals f of x. And we want to sketch on the same set of axes, a graph of y equals f prime of x. So now, remember that the interpretation of the derivative of a function is the slope of the line tangent to the graph of that function. So, we might begin by selecting some random points on the graph. And so, I've selected a bunch of points on the graph. And I've set up these dotted lines to keep our two graphs aligned. Because what I want is the graph of the derivative to show me the derivative at a particular point. So, whatever the graph looks like as it passes through this dotted line, it should correspond to what the derivative is at this particular point. So, what's the derivative? Well, it's the slope of the line tangent to the graph. So, we'll sketch a few tangent lines. It helps if we sketch the tangent lines to be fairly short, because otherwise, they interfere with each other and make it hard to see what those slopes are. So, I'll sketch the tangent line to the graph at these points that I've picked. And at this point, I'm going to start recording information about my derivative, about y equals f prime of x. So, I'll record the sine of the slope of the tangent line. So, here, I'm recording information about the derivative. And here, I have a line whose slope is negative. I have a line whose slope is negative. I have a line whose slope is positive. Slope negative. And so, here's my information about that derivative. So, next, I'm going to plot points whose y-coordinates have the correct sine. So, whatever my point is going to look like, it'll be on this line, and its y-coordinate will be negative. On this line, I want a point whose y-coordinate is negative. On this line, y-coordinate positive. On this line, y-coordinate positive. On this line, y-coordinate negative. And again, since this is just the stick figure version of the graph, we're not going to worry too much about the exact details of where these things are placed, at least not yet, so maybe I have some points that look like this. So, here, I have points with y-coordinate negative. I need y-coordinate positive. That's up here someplace. Y-coordinate negative. That's down here someplace. And for right now, for our stick figure version of the graph, we just want to make sure that these points are located where they should be relative to the x-axis, whether they're below the axis, or above the axis, or below the axis. And now that I have the location of the points, I'll connect the dots to form our sketch of y equals f' of x. And there's our stick figure. Now, sometimes we're actually going to need something a little bit more refined, so here's a different graph, y equals f of x, and this time we'll do the same thing. We'll pick some random points on the graph, we'll draw the tangent line to those points, and we're going to record the sign of the derivative, the sign of the slope of the line tangent to the graph. And so here, all of our tangent lines have positive slope. And this is a little bit problematic, because all that tells us is that our points that we're going to be plotting are somewhere above the x-axis, but we don't know exactly where they are above the x-axis, and we want a little bit more detail than we can get here. So we have to look a little bit more closely at those tangent lines. So we can obtain this additional information by considering the relative magnitude of these values of f' of x. So to begin with, our first point has a positive value. Our first derivative is going to be someplace above the x-axis. Well, the first one is free, so I can just place that point anywhere I want to. But from this point forward, I have to consider what the relative magnitudes of the slopes of the tangent line is going to be. So let's take a look at that. So if I go to the next tangent line, I notice that this tangent line is steeper than this tangent line. Now, be careful with the reasoning here. Both of these have positive slopes, and steeper translates into more positive. So wherever this is, this is some positive number. I want a more positive number here. So that's going to correspond to a point that's a little bit higher up. Now let's take a look at that next tangent line. So this next tangent line is not as steep as this tangent line. So that means that the slope of this is not as positive. It's still positive, but it's just not as positive. And that's going to correspond to a point that's a little bit lower down. And then finally, this last point out here, slope is still positive, but again, it's not as steep as it is over here. So again, it's going to be not as steep, not as positive, and so that'll be a little lower down, importantly, still a positive slope. So that last point is going to be maybe here someplace, and I'll connect the dots and reduce my rough sketch. Now let's take another example. So here's a graph, and we want to sketch the graph of y equals f prime of x. So for future reference, it'll be especially helpful if we can identify places where the derivative either fails to exist, or the derivative is equal to zero. So let's go ahead and start out by finding places where the derivative fails to exist, and these will fall into two types of points. We either get cusps, corners on the graph, or places where the graph is discontinuous. And we can look at the graph and see that it's continuous everywhere. There is this corner right here. There's a cusp here, so the derivative fails to exist, and I'm going to indicate that point, and then I'm going to put a red line there. This is the derivative fails to exist, which means that the graph of our derivative cannot touch this line. This is an electric fence line. Do not touch if the graph of the derivative crosses this line. It is going to be incorrect. The graph of the derivative should not touch that line. The other place we're interested in is finding places where the derivative is zero. Remember, the derivative is the slope of the line tangent to the graph at a point. So derivative zero means that the tangent line is going to be horizontal. And if I look at the graph, I can see that right there at this point, my tangent line is going to be horizontal. And I'll pick a few other points on the graph to help round out the sketch. Because I have this place where the derivative fails to exist, something strange happens here. So I'm going to want to pick a couple of points that are close to it to get a good sense of what's going on there. But the other points can be pretty much anywhere. I have my points where I'm going to be looking at the derivative. I'll draw the tangent lines. Again, don't make them too big. Again, no tangent line at this red line here. And I'll record the slopes of the tangent lines. So positive, positive doesn't exist. Negative, negative zero, positive once again. And so here's our information about the derivatives. I'll locate points in the appropriate places. So here the y-value should be positive, y-value should be negative, positive again. But note that the slope here is less than the slope here. So the y-value should be higher up. In this interval, this line is steeper than this line, which means that the slope of this is more, in this case negative, than the slope of this. So the y-value will be more negative. It'll be farther down. So there's my points. Positive value, positive, but more because it's a steeper slope. Here, negative value, negative, but more because the slope is more negative. Zero, and positive value. I've located my points in the appropriate places, and I'll form my stick figure by connecting the dots. And there's my sketch of the graph of y equals f prime of x. Something strange is going to happen in this interval here, over that is including the place where the derivative fails to exist. But as long as our derivative graph doesn't actually cross this red line, we should be in good shape.