-First you need to know that the difference quotient is known as finding the slope of the tangent line to a graph. On this graph, f(x), we're given a point on the line, which will be (x, f(x)). If we move to a different point, then that will be delta x, or h, for that matter. The new placing of the point, is the total value of x plus h. So at this new point, the values will be (x+h, f(x+h)), which, if you connect to the value (x, f(x)), will create a secant line. Given these points, we can find the slope of the secant line using the formula (y^2-y^1)/(x^2-x^1). We then insert the numbers that correspond with the formula, which will look like this: f(x+h)-f(x)/ x+h-x. If you then simplify, you get the difference quotient: f(x+h)-f(x)/b. Through the process of finding the slope of a secant line, we also find the slope of the tangent line, or the difference quotient.
Source:
1. http://cis.stvincent.edu/carlsond/ma109/DifferenceQuotient_images/IMG0470.JPG
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