1. Explain in detail where the difference quotient comes from.
The difference quotient is derived from a graph in which we are trying to figure out the slope of the tangent line. This slope that we are trying to find of this tangent line is also known as the derivative which is usually shown as f' (x). The difference quotient comes from the point on the graph in which the slope of the tangent line is crossing at. This point is referred to as (x, f(x)). The change of x where another tangent line crosses the graph is known as the change of x (delta x+ x) plus x. The secant line is the place where the line crosses at two points which also helps explain this difference quotient where the second ordered pair is known as (x+deltax, f(x+delta x). After getting these two points you use the slope formula of y2-y1/x2-x1 in order to solve further. When plugging in the values of the ordered pairs you are able to cancel some values and are then left with the difference quotient.
Citation: http://upload.wikimedia.org/wikipedia/commons/8/8c/Derivative.png
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