Question

Calculus help because I am stuck on this question

Answer 1

$y=\left(\dfrac{x^2+2}{x^2-2}\right)^6$

Chain and power rule:

Let $u=\frac{x^2+2}{x^2-2}$. By the chain rule,

$y=u^6\implies y'=6u^5u'$

Quotient rule:

$u=\dfrac{x^2+2}{x^2-2}\implies u'=\dfrac{(x^2-2)(x^2+2)'-(x^2+2)(x^2-2)'}{(x^2-2)^2}$

$u'=\dfrac{2x(x^2-2)-2x(x^2+2)}{(x^2-2)^2}=-\dfrac{8x}{(x^2-2)^2}$

Putting everything together, we have

$y'=6\left(\dfrac{x^2+2}{x^2-2}\right)^5\left(-\dfrac{8x}{(x^2-2)^2}\right)$

$y'=-\dfrac{48x(x^2+2)^5}{(x^2-2)^7}$