In “Local Linearity II”, my post for August 31, 2012, we developed a way of approximating the slope of a function at any point. The slope at *x* = *a* is approximated by

For small values of *h. *

The smaller the better which suggests limits.

The limit of this expression as *h* approaches zero is called the derivative of *f* at *x = a * denoted by :

Now give your students a simple function like *y* = *x*^{2} and give each student a different point in the interval [–4, 4] (include some fractions). Have them calculate the approximate slope and/or the derivative for their point. For each student’s value, plot on a graph the point (their *a*, slope at their *a*). Discuss the results. Guess the equation of the graph.

Of course, the result should look like the line *y* = 2*x*. That is, the derivatives at the various points, taken together, appear to be a function in their own right.

Repeat this exercise with the function *y* = sin(*x*). Guess the equation of the derivative.

We will look at this some more in the next post.