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Calculus/Difference Quotient


Hello, I have a test coming up this week on the difference quotient, and I'm not sure how to solve it when it involves a fraction. One of the examples my professor gave me was to find the difference quotient of f(x)= 1/m
Can you help me solve this?
Thank you in advance.

Take two points, x1 and x2.

It can be seen that f(x1) = 1/m and f(x2) = 1/m.
The difference quotient is (f(x1)-f(x2))/(x1-x2).
In this case, it is (1/m - 1/m)/(x2 - x1).
Since 1/m - 1/m = 0, this is 0/(x2-x1) = 0.

That is the answer, but to look at a more advanced case, consider the following...

If we had f(x) = 1/x, then we would get f(x1) = 1/x1 and f(x2) = 1/x2.
The difference quotient would then be (1/x1 - 1/x2)/(x1-x2).

Combining the fractions in the numerator gives [(x2-x1)/(x1*x2)]/(x1-x2).
This should be rewritten as the quotient of two fractions.
That is the same as [(x2-x1)/(x1*x2)]/[(x1-x2)/1].

Inverting the bottom and multiplying gives  [(x2-x1)/(x1*x2)]*[1/(x1-x2)].
Since (x2-x1)/(x2-x1) = -1, we have -1/(x1*x2).  


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