 Probably the most important aspect of graphing from the derivative occurs when we actually have a graph of the derivative. So here we don't have the function, we don't have the derivative function, but we have a graph of what the derivative will be. So here my y values are the same as my derivative values, and this is a very common situation in measured quantities where it is generally easier to measure the change in some quantity than it is to measure the quantity itself. So here, given a graph of what the derivative looks like, from here can we reconstruct a graph of what the function looks like. So here the thing to notice is that this is the graph of y equals f' of x, which means the y-coordinates are the same as the derivative. So since one of the things we'd like to find is where the derivative is zero, well the derivative is the y-value, is the y-coordinate. So where is the derivative zero? That's any place where the y-values are zero, and that means any point along the axis, which has a y-coordinate of zero. So those are important points. We also want to know the sign of the derivative at places in between. So the sign of the derivative, the derivative is the y-value, which means that anytime the y-values are negative, the derivative is negative, anytime the y-values are positive, the derivative is positive. So here the y-values the derivative, we are positive if we're above the axis. The y-values the derivative were negative if we're below the axis. The y-values the derivative were positive if we're above the axis. And so now I know the signs of the derivative. Now we'd also like to find some information about the second derivative. So remember the second derivative is the derivative of the derivative. So since my graph is the derivative itself, the derivative of this is going to be my second derivative. Well, remember the derivative of any function that produces a graph is going to correspond to the slope of the line tangent to the graph of whatever our function is. So if I want to find the second derivative, I want to look at the slope of the line's tangent to the graph of the derivative. So one of the ways we can do that is we can look for points where the tangent lines are going to be horizontal. So here the tangent line will be horizontal, here the tangent line will be horizontal, and then look for the slopes in between those places. So all along here our slope of the line tangent to the graph, our tangent lines are going to be pointing this way, pointing downwards. So all along here our derivative will be negative, all along here our tangent lines will have positive slope, and then along this last section our tangent lines will have negative slope. So the slope of the line tangent to the graph of this will have slope negative, positive, negative. Well, the slope of the line tangent to the graph is the same as the derivative, well that's the derivative of the first derivative and that's our second derivative. So we've identified where the second derivative is negative, is zero, is positive, is zero, negative, and again it's helpful to find these signs of the derivative and the second derivative in the same location. So we'll fill in those places. Again, you can only change signs of the zero. So if you're negative here, you're negative everywhere in this interval. If you're positive, you're everywhere positive in the interval, and so on. So here's our signs of the derivative and the second derivative. So now let's go through the function. So the first portion has second derivative negative, which means that the graph is going to be concave down, and the portion that we're looking at is going to have first derivative positive or zero or negative depending on where we are. In fact, if we look at what our derivative is, our derivative goes from positive to zero to negative, which means that we run the entire length of this from positive slope to zero slope to negative slope. So that first portion of the graph, where concavity is down, looks exactly like all of that. So we'll just translate all of that over to there. The second portion, I have second derivative positive, so that means the second portion will look like part or all of a concave up graph, and in this case we are looking at the slope, the first derivative, changing from negative to zero to positive. Well here's slope negative, slope zero, slope positive, so we're going to include all of this in our second portion of the graph. The last portion of the graph, second derivative is negative, which means that we're looking at something that's concave down, but our first derivative is going to be positive, which means that for this second portion of the graph, we're only looking at the section of a concave down graph with positive slope, and that means just this first part of the graph. So our last bit of the graph is going to look like that, and there's the complete sketch of our figure. Now here's an important thing, the shape of a graph can be very very different even though the derivative looks almost the same. So here's a variation, so here's our derivative graph this time, and note that this derivative graph and our derivative graph that we just did look very similar, but as we'll see the graphs that they produce are going to be radically different. So again let's take a look at that. So here we want to see what happens if we look at this graph, and again the important things about the derivative, the derivative, which is the y value, are whether the derivative is the y value, positive, negative, or zero. Well since all of this graph is above the x-axis, that means at every point on this graph the y values are positive. The y values are positive, the derivative is positive. So again I'll take my note, derivative, positive, everywhere here. There's no place where we change sides. The next thing we can look at is the second derivative that's going to be the slope of the line tangent to the graph of y equals f prime of x, well here's our graph, and we are interested in knowing where that tangent line has slope zero, or positive slope, or negative slope, and we'll take a look at those places, and we find here and here the tangent line is horizontal, the tangent line has slope zero, all along here tangent line has negative slope, tangent line has positive slope, tangent line has negative slope, and that is the derivative of f prime of x, second derivative. So we have the signs of the second derivative, and again first derivative is always positive, and now we have both the sign of the first and second derivative, so we can begin to graph. So for the first part of the graph, second derivative negative, which means we look like part of a concave down graph, first derivative positive, which means we look like the first part of a concave down graph. So we'll sketch that in. Second section, derivative, second derivative is positive, so we look like part of a concave up graph, first derivative positive, so we look like this portion of a concave up graph. Last section, second derivative is negative, first derivative is positive, so we're going to look like the rising part of a concave down graph. Second derivative negative, so we're concave down, first derivative positive, look like that last bit, and so there's our sketch of the graph of y equals f of x, and so again notice that our graph looks very different from the previous one. So this is the graph where our derivative is entirely above the x-axis, but then if our derivative is below, partly below and partly above the x-axis, almost the same graph in appearance, the actual graph of the function looks very different.