I think the use of Algebra Xpressor will provide meaningful knowledge about questions raised above by comparing a mother function and its differentiated fuction visually in a scene. Next are an example of rational function . The graph can be easily drawn by Algebra Xprossor, but, the main focus is how to use it to make students understanding about relationship between a function and its differential function . Let's look at the function graph first. There are three distinct parts of a graph.
Students can draw an arbitrary tangent line on a point of the graph.
The slopes of all the tangent lines of the graph will explain another function. Since the sines of slopes are changing along the graph. Algebra Xpressor can draw the differentiated function if students can calculate the derivative function of the mother function. Following is the graph of the derivative function . Students will have a chance to discuss about the case of value when the denominator of a rational function is zero.