How to Sketch the Derivative Graph Given the Graph of the Function
If we are given a graph and want to sketch the graph of the derivative, we just have to remember that the derivative is the slope of tangent line. We can form a TTable where the coordinates are (x, f '(x)). Example Consider the graph shown below. Sketch the graph of its derivative.
We will just eyeball the slope of the tangent line. Notice that this slope is 0 for x = 2.5 and x = 1. Now draw a TTable of derivatives. We start with x = 5. The graph is almost horizontal but has a very slight downward slope. Hence the derivative is approximately 0.2. At x = 4, the graph is going downward but much steeper. The slope of this tangent line is approximately 2. At x = 3, the function is not continuous, hence it is undefined. We write "DNE" to mean "Does Not Exist."
Now use the TTable to plot the points. (5, 0.2), (4, 2), etc. Then connect the points with a curve. The graph of the derivative is shown in green.
