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.
On
Most people screw up the Quotient Rule because they forget it. Quite actually, the quotient rule is just a corollary of the product rule (ie, the quotient rule is just a special case of the product rule)
The product rule states that that for two functions $u$ and $v_0$,
$$(uv_0)'= uv_0' + u'v_0$$
Let $$v_0 = \frac{1}{v} \implies v_0' = -v'/v^2$$
So, $$ \left(\frac{u}{v}\right)' = u\cdot\frac{-v'}{v^2} + u'\cdot\frac{1}{v} = \frac{-uv'}{v^2} + \frac{u'v}{v^2} = \frac{u'v - uv'}{v^2}$$ Hopefully, that was helpful.
$$\frac{a}{b}-\frac{a}{c} \neq \frac{a-a}{b}$$ but $$\frac{a}{b}-\frac{a}{c}=\frac{ac-ab}{bc}$$
Notice that you have different denominators.