Unit V - BQ #7: Where does the difference quotient come from?

WHERE DOES THE DIFFERENCE QUOTIENT COME FROM?

First of all, a function of any type is needed to figure out where the difference quotient comes from. Take a secant line (a line that touches the function twice) and use the points. The first point is x units on the x-axis and f(x) units in the y-axis, therefore being a coordinate pair of (x, f(x)). The second point is taken, however it has an h units difference from the first and therefore being a coordinate pair of (x+h, (f(x+h)). 

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Plug the two points into the the slope formula [m = (y2-y1)/(x2-x1)] to find the slope.  After the appropriate information has been correctly plugged in, there is some simplification/combining-like-terms that occurs at the denominator. Once the simplification has been complete, the equation left  is [(f(x+h)-f(x)]/h, which is known as the difference quotient. This is used to find the slope of a tangent line, which, unlike a secant line, is a line that only touched a function once.