Name: Milena
Who is asking: Parent
Level: Secondary
Question:
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
Cheers
Walter Whiteley
York University
