Where does the difference quotient come from and what do you know about it?

The difference quotient has a nickname called the derivative which is derived from the slope formula of (Y^2-y^1)/(x^2-x^1). The reason why we use the slope formula is because the derivative is the slope of all the tangent lines. A tangent line touches the graph one time. Looking at our formula for the difference quotient is f(x+h)-f(x)/h. We have two points that we use in our slope formula. Our first x point is simply just x. While our first y value is f(x) As we move along the x axis we have our letter H which can be referred to as delta x. Our second points are used from the original ones but including h because of the shift. So our points would be (x+h) and then our second point would be f(x+h) We then plug this into our slope formula and get f(x+h)-f(x)/f(x+h)-x we then simply our x values in the denominator and whats left is our difference quotient f(x+h)-f(x)/h.

By looking at the picture above our delta x or our h has to be moved closer to be able to be touching which is the secant line that touches two points. In the SSS videos our delta x has to be moved closer and closer in order for the points to be directly touching.

Reference:

http://0.tqn.com/d/create/1/0/9/p/C/-/slopeformula.jpg

http://cis.stvincent.edu/carlsond/ma109/DifferenceQuotient_images/IMG0470.JPG

http://images.tutorvista.com/cms/images/39/difference-quotient-formula.png

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