BQ#7 - Unit V

Explain in detail where the formula for the difference quotient comes from now that you know! Include all appropriate terminology (secant line, tangent line,h/delta x, etc).

In this unit, we have been utilizing the difference quotient to solve derivatives. This raises the question: "what is a derivative?" Well, the derivative is the slope of the tangent line to a graph (the graph is most commonly referred to as "f(x)" while the derivative is noted as "f'(x)." To better understand what a tangent line is, we should look into secant lines. A secant line is a line that passes through a graph at two points; however, if you were to move those two points closer and closer together, until they meet, you would receive a tangent line - a line that touches the graph only once. In order for a secant line to turn into a tangent line, we must alter the slope. 

http://cis.stvincent.edu/carlsond/ma109/diffquot.html

In the diagram above, we see two lines. Ultimately, our goal is to alter the red secant line in order to create a tangent line. Remember, this involves using slopes! This is where the slope formula () comes into play. However, for this to work, we need points to plug into the formula. Looking at the picture above, we notice that our two points will be (x, f(x)) and (x+h, f(x+h)) {"h" in this equation is solely the difference in space between the two points}. WOO! We now have our two points to plug into the slope formula. The work for this is shown below!

In the picture above, we can see the two points plugged into the formula for a slope. Nothing in the numerator can cancel out. However, we CAN cancel out the "x" on the bottom! This leaves us with our difference quotient!!!! 

Sources: http://cis.stvincent.edu/carlsond/ma109/diffquot.html
http://math.about.com/od/allaboutslope/ss/Find-Slope-With-Formula-JW.htm