 Since the derivative is a slope of the line tangent to the graph, we can sketch the graph of the derivative from the graph of the function. It helps if we consider the following. The value of the derivative, or any number, can be separated into two parts. The sine, positive or negative, and the magnitude. To avoid getting lost in the details, we'll focus on the sine first. So we might try to find the sine of the derivative at the points indicated, and so we have our graph. And so if you remember only one thing about the derivative, you should have paid more attention. But the one thing you should remember is that the derivative is a slope of the line tangent to the graph. So let's draw our tangent lines at A, B and C. At A, the tangent line slopes upward, so the slope of the tangent line is positive. So the derivative is positive. At B, the tangent line slopes downward. So the slope of the tangent line is negative, and so the derivative is negative. At C, we see the tangent line is horizontal, and so the tangent line has slope zero, and the derivative is equal to zero. Suppose we have the graph of a function. Let's try and draw on the same set of the axis the graph of the derivative. Now it's important to understand what this means, since both graphs are on the same set of axes. A point on the graph of y equals f of x will correspond to a point with the same x-coordinate on the graph of y equals f prime of x. The other thing to remember is that if y is greater than zero, the point is above the x-axis. If y is less than zero, the point is below the x-axis. And if y is equal to zero, the point is on the x-axis. We can begin by picking a few random points on the graph, then draw the tangent lines. It may help to pick one point to start with. So let's start with the starting point here A. At A, we'll draw the tangent line, and we see the tangent line has a positive slope. So the derivative will be positive. But since we're graphing y equals f prime of x, then y will be positive. And so the corresponding point on the graph of y equals f prime of x will be above the x-axis. Now if we follow the graph, we see that the slope of the tangent line will continue to be positive until we get to B. At B, the tangent line is horizontal. So the slope is zero. So the point on the graph of y equals f prime of x will be on the x-axis. And so we'll draw the point. If we continue onward, at C, our tangent line will slope downward. So the slope is negative. So f prime of x is negative. So y is negative. And so the corresponding point is below the x-axis. And if we continue on to D, our tangent line will be horizontal. So its slope is zero. So f prime of x is zero. Equals means replaceable, so y is zero. And so the point is on the x-axis. And if we continue to E, our tangent line will slope upward. So its slope is positive. And so f prime of x is positive. And so the corresponding point is above the x-axis. Now we just have a couple of points on the graph of y equals f prime of x. And for now, let's just connect the dots. And that gives us the sketch of the graph of y equals f prime of x.