 The derivative is the slope of the tangent line, and one of the things that that means is if you actually have the graph of a function, you can actually get a pretty good idea of what the derivative is going to look like. So if I have a graph of y equals f of x, I can actually try and sketch the derivative graph without having to do the differentiation. And this is particularly good, because certainly as we start, we're not going to know a lot of derivatives off hat. And part of the reason is that a little bit later on, we want to be able to go forward from the graph of y equals f of x to the graph of the derivative y equals f prime of x, and that's mainly because a little bit later on, we're going to want to go from the graph of the derivative back to the graph of the original function. So of course we want to start with a perfect graph, but most of us aren't that good, so we'll start with a stick figure, you know, call that a rough sketch of the graph of y equals f prime of x. And a little bit later on, we'll refine that to try and get a better sketch of y equals f prime of x. Now what do we need? Well, there's three important components that we want. We want to identify where the graph of y equals f prime of x does not exist, where there are gaps in the graph. Those are places where the derivative fails to exist. Where the graph of y equals f prime of x is above the x-axis, that is to say our y values are positive, so f prime is positive, and then where the graph of y equals f prime of x is on the axis, which is to say if you're on the axis, your y value is zero, where is the derivative zero, and lastly, we want to find where the graph of y equals f prime of x is below the x-axis. The y values below the x-axis, y is negative, derivative is negative. And once we can identify these points, we can connect the dots, and then later on we'll refine our sketch of the graph. How does that work? Well, let's consider a graph. So here's a graph, y equals f of x, and we want to sketch on the same set of axes a graph of y equals the derivative of x. Now the derivative corresponds to the slope of the line tangent to the graph, so let's pick a couple of random points on the graph. So I've selected a bunch of points here. I've drawn these vertical lines to indicate where we are located, because what I would like to do is I'd like my derivatives to correspond to the same locations. So at this x-value, my derivative is whatever it is. At this x-value, my derivative is slope of line tangent to graph. At this x-value, slope of line tangent to graph, and so on. So I'll sketch the tangent lines to the graphs at the points that I've chosen. And then after I've sketched the tangent lines, the thing that I want to do, the derivative is the slope of the line tangent to the graph. So I want to record the sign of the slope of the tangent line. So let's go ahead and take a look at them. At this point here, the tangent line slopes downward, derivative is negative. Tangent line slopes downward, derivative is negative. Tangent line slopes upward, derivative is positive. Tangent line slopes upward, derivative is positive. line slopes downward, derivative is negative. Now because I'm just making a rough sketch, because I'm just making a rough sketch, the first thing I would like to get is to make sure that I'm actually in the right part of the graph, the right part of the plane. So here I want to just plot points whose y-coordinates have the right sign. So I'll plot a few points. So here I want y-coordinate negative, that's below the axis some place, and I want to line it up on this vertical line. Here, again, y-coordinate is negative, so that's going to be some place below the axis, some place on this line, how about right there. There's a reason in our more refined sketch, we'll see why we place this one a little bit higher up, but for right now it doesn't really matter too much. So here we want y-coordinate positive, that's above the axis some place on this line, y-coordinate positive, above the axis some place on this line, y-coordinate negative, below the axis some place on this line, and there's a bunch of points that at least have the right sign, and I'll connect the dots to form our stick figure, again our sketch of the graph of y equals f prime of x. Now later on we'll actually refine this graph, but for now this is actually not a bad starting point. Let's take a look at another one. Here's y equals f of x, a much more complex graph, and something we might do to make our lives easier. Rather than just picking random points, we could start by picking significant points. So the important points are the places where the derivative does not exist, and the derivative is zero. And part of the reason is that the other points that matter, where the derivative is positive, where the derivative is negative, the only way we can go from negative to positive, or vice versa, is we either pass through a place where the derivative is zero, or the derivative fails to exist. So let's identify where those points are. So here we have a corner, and we know that at corner points, derivatives does not exist, so that's an important point. And then imagine what those tangent lines look like, so along here derivative is positive, along here derivative is negative, derivative positive right here. If I draw the tangent line through this point, the slope of that line, it's horizontal, the slope of the line is going to be zero. So here I have a place where the derivative does not exist, here I have a place where the derivative is zero, and in between them, I have places where the derivative doesn't change signs. And again, just to get some idea of what those are, we might pick a couple of intermediate points and draw the tangent lines through those points. And again, since I want the graph of y equals f prime of x, I want to record the sine of f prime of x, and so I'm looking at the slope of the tangent line. Here the slope is positive, here the tangent line doesn't exist, here tangent line slope is negative, tangent line slope is zero, tangent line slope is positive. Next, I want to plot a couple of points whose y-coordinates have the correct sine. So again, I want my y-coordinate to be positive, some place above the axis doesn't exist, I want my y-coordinate to be negative, some place below the axis, y-coordinate zero right on the axis, y-coordinate positive, some place above the axis, and now I can connect the dots. With one important reminder along this line, the derivative doesn't exist. So I need to make sure that my graph has a hole where it doesn't exist. And so here's my sketch of the graph of y equals f prime of x. Now, from time to time, we'll run into a graph that makes this a little bit more complicated, and more generally, if we want to get more than just a stick figure sketch, we're going to have to think a little bit more about what the derivative graph looks like. So for example, here's another function, and here's the graph of that function. And I might select some random points and draw the tangent lines to those points, and then record the signs of the derivatives. And here's the thing I notice here. Slow positive, slow positive, tangent line slope positive, tangent line slope positive. And the problem here is if I try and graph y equals f prime of x, every point that I plot is some place above the x-axis. And I don't have a really good sense of what that looks like. So here's a couple of points above the x-axis. So maybe my derivative looks like that. Here's a bunch of points along the x-axis. Maybe my graph looks like that. And then maybe I can do other things that all have points above the x-axis. So it's a little difficult to get a good sense of what the derivative graph looks like. And the way we can do that is by considering the relative magnitude of the derivatives. And the key question here is the slope greater, a larger positive number, or less a smaller number. And for that, we can look at the steepness of those tangent lines. So let's start. Our first point is going to be y-coordinate positive. So we're someplace above the x-axis. So we'll go ahead and plot our point there. First point is free. And the thing that we have to ask is not only the second point is above the x-axis, but how does the slope of the second point compare to the slope of the first point? And if we look at it, the slope is deeper, the second line is steeper, and the slope is positive. So whatever the slope is for this, the slope of this line is going to be greater, more positive, so that next point is going to be higher up. Well, let's continue that analysis. If I go to this next point, here's the tangent line. Here's the tangent line of the point I'm at. And this line is deeper than this line. So the slope of this line is greater than the slope of that line. So the derivative will also be greater. And so that means the next point should actually be a little bit lower down to reflect the fact that the slope is not as great as it had been. And then if I go to that last point, I again have a positive slope, but this is even smaller than this positive slope. So my corresponding point should be even lower down. And now I play connect the dots. And there's my sketch of y equals f prime of x.