Throughout these 10 months, we have learned to memorize and be able to use the difference quotient. It has always been just another song memorized to me, however I know understand how the difference quotient formula is derived. The different quotient is derived because the function is "touched" twice, which is called a secant line, as shown below in orange. A tangent line is when the function is only "touched" once, as shown below in the pink line.
http://www.mathsisfun.com/definitions/secant-line-.html
We know that the limit of the formula is whenever h=0, and that only happens when there is a tangent line. The slope of the tangent line is usually written as f(x). To derive the difference quotient, we can plug x in for the difference quotient and solve through for whenever there is a tangent line.
