As I stated in the intruction of my project for EMT 670 students in Korean high school are expected to solve complicated function problems without help of any technology. So the knowledge about differential calculus is as important as solving the problems themselves. Let us guess that a rational function is given. Students can get a lot of information of the function with the differentiated function. Students first will graph the simple figure of the function, and usually mark the roots of function, if possibly calculated, and some points where the differentiatial coefficients are zero. I think every knowledge from the process should be understood conceptually by students. "Why do they try to find the roots of a function?" and "What does it mean for the students to have 0 as differential coefficients?". 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 . One thing that students have to notice is that the actual graph does not have a value when x=2 and x=-2, but, Algebra Xpressor shows the two lines x=2 and x=-2. Students will have a chance to discuss about the case of value when the denominator of a rational function is zero.

Look at the following graphs and see if we can find significant information by comparison.

Red is a function , and green is its differentiated function . Look at the region with 4<x<10 of two graphs. What does green say? If a student draw a tangent line on a point in 4<x<10, then the tangent line will have positive slopes and the slopes will be increasing. Green graph is showing the exact movement of the slopes of tangent lines.
What if I draw a tangent line with x=1? Green graphs says that the value is negative so the tangent line will be a linear function with a negative slope. Same kind of logic can keep going on for x<-3.
If you would have drawn the differentiated function first, then you could have understand that you would see three different parts of a function .