Find the difference quotient f(x+h) - f(x) /h , where

f(x)= 7 / x-3

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f(x+h) - f(x)= 7/(x+h-3) - 7/(x-3)=7(x-3)/[(x+h-3)(x-3)] - 7(x+h-3)/[(x+h-3)(x-3)]

Since these now have common denominators distribute the seven in the top of the fraction and combine.

7x-21-7x-7h+21/[(x+h-3)(x-3)]= -7h/[(x+h-3)(x-3)]

Now divide this by h as the difference quotient tells one to do. This will cancel with the h in the numerator.

The answer is now -7/[(x+h-3)(x-3)]. As the h goes to 0 you just plug in 0 for h leaving -7/[(x-3)(x-3)] or -7/(x-3)^2.

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