The Derivative Function
Since the derivative of a function f at any value x 0 where the function is differentiable is just a number, f'(x 0) , 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.