**This song rather nicely lays out
the mathematical definition of the derivative of a function
***y*(*x*). Most fundamentally, the derivative
of *y*(*x*), which is written
and pronounced "dy dx", is defined as the slope
of the graph of *y* vs. *x*. In the figure,
at the point *x*_{0}, the slope of the curve
is equal to the slope of the green line which is tangent to
the curve. We can very roughly approximate this slope by computing
the slope of a line drawn between the points on the curve
at *x*_{0} (pronounced "x nought")
and at *
*(pronounced "x nought plus Delta-x"), as shown
by the red line. The slope of this line is the "rise
over the run", in other words the slope is
(the change in *y*) divided by
(the change in x). If we make
smaller and smaller, then
also gets smaller, and the slope of the red line gets closer
and closer to the slope of the green line. In the ultimate
"limit" that
becomes infinitesimally small (in the song "send delta-*x*
to zero"), the two slopes are equal.

**--WFS 9/1/05**