<s>[INST] Please explain the Quotient Rule I am currently working on an equation but I'm having a hard time understanding how to get the answer.
the answer is ${(x^2-4)(x^2+4)(2x+8)-(x^2+8x-4)(4x^3)\over (x^2-4)^2(x^2+4)^2}$
The equation is $f(x)= {x\over x^2-4}-{x-1\over x^2+4}$
When I apply the quotient rule i get $f'(x)= {(1)(x^2-4)-(2x)(1)\over (x^2-4)^2}-{(1)(x^2+4)-(2x)(1)\over (x^2+4)^2}$ but it cancels each other out.  I can't figure out how they had gotten the answer.
 [/INST] $$\frac{a}{b}-\frac{a}{c} \neq \frac{a-a}{b}$$ but $$\frac{a}{b}-\frac{a}{c}=\frac{ac-ab}{bc}$$
Notice that you have different denominators.
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