Name: Milena
Who is asking: Parent
Level: Secondary

This question has been nagging me for sometime now.

Is there a way of finding out the derivative of a function, just by looking at the graph of it?

Hi Milena

The simple answer is YES, provided that the scales on the x and y axis are the same. Of course, you are going to find the GRAPH of the derivative, not the FORMULA for the derivative!
   I normally give exercises to students where I sketch the graph and they setch a graph of the derivative.
   Consider a graph. Take a horizontal ruler. You can slide it up and down and recognize (approximately) where the tangent line is parallel to the horizontal. These are the points where the derivative is 0.
   Take a ruler at a 45 degree angle (tangent = slope is 1). Slide it along and detect the points where it is tangent. These are the points where the derivative is 1. You can easilly read off the points where the derivative is positive (the function is going up or increasing). Similarly, you can read off where the graph is going down. These are the points where the derivative is negative.
   You already have a lot of information for a graph of the derivative. More points can be sketched in as needed.
   This is best for certain QUALITATIVE information - which is often what we want.
   If you enjoy this graphic exercise - try the reverse: draw the graph of the derivative and then sketch the graph of the the original. Sure you have to choose some arbitrary height to start at - but the rest is determined. This is the visual/graphical way of 'doing integration'.
   There are illustrations of this visual process in some videos and, I believe, in some demonstrations on the web (e.g in Geometer's Sketchpad demonstrations from the Math Forum

Walter Whiteley
York University


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