BQ #7: Explain in detail where the formula for the different quotient comes from now that you know! Include all appropriate terminology (secant line, tangent line, h/delta x, etc).

We have known about the difference quotient for a long time now; in fact, it is often included in our tests. However, we really didn't know where it came from until we reached the last unit- Unit V! So let's take a closer look at how the difference quotient is derived.
Based on the first few concepts, we know that the difference quotient is only a part of the process when trying to find the derivative (the slope of tangent lines).

http://t3.gstatic.com/images?q=tbn:ANd9GcQx-jY-1cjcu_PSC9gGtoDW4qMYafVwaqWw8Qt5qV_gQpIKlz5w:cis.stvincent.edu/carlsond/ma109/DifferenceQuotient_images/IMG0470.JPG

Notice that the image on the left shows us two  main points. The first point is [x, f(x)]because the first value on our scale was "x". Since the first value under the x-axis is "x" that means that the value for y in that point is f(x).  There is another point, but does not have exact digits either. Therefore, we simply add the "h" (the distance between the two points) to the already known "x", so we can get "x+h". Then as we did before, the y-axis will be f of the x-axis, which turns out to be f(x+h).

http://t2.gstatic.com/images?q=tbn:ANd9GcSmctLWns4VWjJOvK672RQnfSPeu-73rvYghAvuGT_HhHTEhVRY3w:education-portal.com/cimages/multimages/16/Slope_formula_2.png

We continue to find the difference quotient by using the slope formula as shown above. After plugging in the values based on the slope formula we get [ f(x+h)-f(x)/ x+h-x]. Since the "x" in the denominator cancel out, we only have to simplify it to get [f(x+h) -f(x)/ h]....the difference quotient!!
Like I mentioned before, the difference quotient can be used to find the slope of a tangent line, which only touches the function once. We do not have to rely on the secant line because it touches the function twice. As a matter of fact, the reason why we try to find the limit as h approaches 0 is because we want "h" to be as close as possible to 0 so that both points can meet at one point instead of two like the secant line).