Math Has Serious Problems.: BQ#7: Unit V - Derivatives and the Area Problem

1. Explain in detail where the formula for the difference quotient comes from. Include all appropriate terminology.

First of all, the equation of the difference quotient is:

It is used to find the slopes of the graph. To derive it, look at the picture below. The first intercepted point is (x, f(x)) and the second is (x+h, f(x+h)). On a side note, the red line is called a secant because it's intercepted at two points and a tangent line is when it intercepts only at one point. Using the two points, plug them into the slope formula: (Y2-Y1)/(X2-X1). Once you have substituted these points, you'll see that the x in the denominator can cancel, leaving an h in the denominator. Once done simplifying, you'll see that it looks exactly like the difference quotient. The difference quotient finds the slope of a tangent line to a graph. This is also known as the derivative.