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 T-Table where the coordinates are (x, f '(x)).
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 T-Table 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 T-Table 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.