Simplifying Second Derivatives 
I can't seem to figure out how my professor simplified this second derivative. Any help is much appreciated. I'm having trouble simplifying the second derivatives of most problems so step by step instructions would be awesome.

Notice that each term in the numerator has a common factor of $1-x^2$ so cancelling that you are left with
$$\begin{align}\frac{(6x)(1-x^2)^2-3(1+x^2)(2)(1-x^2)(-2x)}{(1-x^2)^4} &= \frac{6x(1-x^2)-3(1+x^2)(2)(-2x)}{(1-x^2)^3} \\ &= \frac{6x(1-x^2)+12x(1+x^2)}{(1-x^2)^3} \\ &= \frac{6x-6x^3+12x+12x^3}{(1-x^2)^3} \\ &= \frac{18x+6x^3}{(1-x^2)^3} \\ &= \frac{6x(3+x^2)}{(1-x^2)^3}\end{align}$$
