Education Technology

The Relationship between the Graph of a Function and the Graph of its Derivative

Published on 06/09/2008

Activity Overview

In this Derive™ activity, students relate the behavior of the graph of an exponential function to the signs of its first two derivatives. They realize that if the signs of the first two derivatives over a given interval are known, a fairly accurate picture of the graph of the function can be obtained.

Before the Activity

  • See the attached DFW file for detailed instructions for this activity
  • Print pages from the attached DFW file for your class
  • During the Activity

    Distribute the pages to the class.

    Follow the Activity Procedures:

  • Set up a 2D plot window
  • Set the plot range and graph an exponential function
  • Insert annotations where the curve rises or falls
  • Use the Derive™ software to differentiate the function, and graph the result
  • Label where the graph is positive or negative
  • Test the sign of the derivative of the function with the help of Derive's 'If' syntax


  • Evaluate the second derivative of the function, and graph it along with the function in the plot window
  • Label the function where it is concave or convex
  • Label the graph of the second derivative where it is positive or negative
  • Check if the regions on each graph coincide


  • Combine the first and second derivative tests using a logical AND operator
  • Record the intervals on the x-axis where the graphs are positive or negative
  • Observe the points where the derivatives are zero
  • After the Activity

    Students answer questions listed on the activity sheet.

    Review student results:

  • As a class, discuss questions that appeared to be more challenging
  • Re-teach concepts as necessary