[tex]y=\left(\dfrac{x^2+2}{x^2-2}\right)^6[/tex]
Chain and power rule:
Let [tex]u=\frac{x^2+2}{x^2-2}[/tex]. By the chain rule,
[tex]y=u^6\implies y'=6u^5u'[/tex]
Quotient rule:
[tex]u=\dfrac{x^2+2}{x^2-2}\implies u'=\dfrac{(x^2-2)(x^2+2)'-(x^2+2)(x^2-2)'}{(x^2-2)^2}[/tex]
[tex]u'=\dfrac{2x(x^2-2)-2x(x^2+2)}{(x^2-2)^2}=-\dfrac{8x}{(x^2-2)^2}[/tex]
Putting everything together, we have
[tex]y'=6\left(\dfrac{x^2+2}{x^2-2}\right)^5\left(-\dfrac{8x}{(x^2-2)^2}\right)[/tex]
[tex]y'=-\dfrac{48x(x^2+2)^5}{(x^2-2)^7}[/tex]
