Since the derivative of a function f at any value x0 where the function is
differentiable is just a number, f'(x0), we may define the derivative function,
denoted f'(x) or
(x), to be the function that maps a value x where
f is differentiable to the derivative of f at x. Since f' is a function in
its own right, we can graph it, just like any other function.
For example, let us calculate the derivative function (also called simply the
"derivative") of f (x) = 5x. Since the graph of this function is a line with slope
5, we might guess that the derivative will equal 5 at every point. We verify this
with the formula for the
derivative of f at a point x:
Indeed, f'(x) = 5, a constant function.