The derivative of a function is another function. Geometrically, in terms of the graph, the instantaneous rate of change corresponds to the slope of the tangent line. Several on-line animations and applets are available to illustrate how this varies as we move along the curve, and also to show how it can be found as a limit of secant line slopes . A popular choice is to use instead the expression 'delta x' to emphasize that it referrs to a d ifference of x values. Another notation that is often used is due to Leibnitz, the other main inventor of Calculus.
Some modern texts use the notation Df rather than f ' .
If you have come across any good web-based discussions of the definition of the derivative, please do let us know and we will add them here.