How do you graph the derivative of a function when you are given the graph of the function?

Mar 7, 2015

You do this by looking at how the slope of the lines tangent to the graph change as the value of $x$ (the independent variable) changes.

This works, because the derivative gives us a formula (a function) for finding the slopes of tangent lines for various values of $x$.

Consider the following graph of a function below.
Notice that as we look from left to right, the tangents on the left have large positive slope.
The slope decreases (the tangents are more nearly horizontal) as we pass x=0 (the $y$-axis) and get close to the high point around x=0.5. The $y$ value here is a bit more than 2 and it is called a local or a relative maximum value).
Continuing our rightward journey, the slopes of the tangent lines become negative and decrease to about x=1.5 after which point the tangents once again get flatter (closer to horizontal). We arrive a a local minimum value when we reach 2.3 or so and continue into a part of the graph where the tangent lines have positive slope.
graph{x^3-4x^2+2x+2 [-3.19, 7.91, -2.93, 2.62]}
Here it the graph of the derivative of the function above:
graph{3x^2-8x+2 [-3.416, 6.45, -3.848, 1.087]}

Notice that the local extreme values for the function occur at the same $x$ values that make the derivative $0$. (The value of the function is $0$ at the $x$-intercept of the graph.

Should you wish to try this with a graphing utility and other functions: try:
$f \left(x\right) = {x}^{5} - {x}^{4} - 3 {x}^{3} + {x}^{2} - x$
and
$f ' \left(x\right) = 5 {x}^{4} - 4 {x}^{3} - 9 {x}^{2} + 2 x - 1$.