# The Derivative I

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

$\displaystyle \frac{f\left( a+h \right)-f\left( a \right)}{h}$

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 ${f}'\left( a \right)$:

$\displaystyle \ {f}'\left( a \right)=\underset{h\to 0}{\mathop{\lim }}\,\frac{f\left( a+h \right)-f\left( a \right)}{h}$

Now give your students a simple function like y = x2 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 = 2x.  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.

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