In the process of finding the difference quotient we must understand slope and how a slope is found. The slope will give us two points for a line that will go through the function, that line is known as the secant line.
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| (http://clas.sa.ucsb.edu/staff/lee/Secant%20and%20Tangent%20lines.gif) |
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| (http://0.tqn.com/d/create/1/0/9/p/C/-/slopeformula.jpg) |
We have no points when finding the derivative of the difference quotient, we only have the x and y axis, although we should plot two points if we want to find the derivative of the difference quotient. X1 would be labeled as x and a value called h is added to reach x2 on the x axis making x2 as x+h. To find the y values we just plug in our x values into the function f(x) and our point for y1 would be f(x) and y2 would be f(x+h). Finally we plug in these points to the slope formula (y2-y1)/(x2-x1), so we would have f(x+h)-f(x)/x+h-x and the x's in the denominator would cancel leaving us with f(x+h)-f(x)/h.
Resouces
http://0.tqn.com/d/create/1/0/9/p/C/-/slopeformula.jpg
http://clas.sa.ucsb.edu/staff/lee/Secant%20and%20Tangent%20lines.gif
