Derivative Viewer
Simultaneously study a function and its derivative
Enter an equation describing a curve in the text field below, (in the same way as for the Function Viewer activity). You can scale the plots using the sliders, and pan by dragging the background. The curve will be plotted in the upper plot, and its derivative (i.e. its slope curve in the lower plot.

Click on the curve and drag to view the slope at any point, both as a tangent line in the upper plot, and a point in the lower plot. See how the y-coordinate of the point in the lower plot corresponds to the slope of the tangent in the upper plot. Click on the background to hide the tangent and points.

This activity allows you to interactively explore the nature of a function's derivative.

Function:      

x scale:       y scale:   

About the activity

There are many uses of calculus—both pure and applied, but generally in the applied mathematics case, the curve or function being studied describes how some quantity (the dependent variable) changes in response to changes in some other quantity (the independent variable).

Natural questions to ask include:

  • When is the function increasing?
  • When is the function decreasing?
  • When is the change fastest?
  • When is it stationary (i.e. not changing at all)?
  • Where do the maximum and minimum values occur?
For all of these questions, considering the derivative curve provides answers. Try plotting various functions and see if you can work out exactly how the derivative answers these questions. Some example curves you might want to try are:
x2, x3, x − x3, sin(x) and abs(x).

The significance of calculus and its ability to find and analyse the the important points of a continuous function is nicely captured in the following statement by Leonhard Euler, the greatest mathematician of the years following Newton, and indeed one of the greatest ever:

Nothing takes place in the world whose meaning is not that of some maximum or minimum.
This is the power of mathematics—it helps bring meaning to the world. And that's truly wonderful.