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

##### 1 Answer

You do this by looking at how the slope of the lines tangent to the graph change as the value of

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

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

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

Should you wish to try this with a graphing utility and other functions: try:

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