Question

Calculus help because I am stuck on this question

Answer 1

`y=\left((x^2+2)/(x^2-2)\right)^6`

Chain and power rule:

Let
`u=(x^2+2)/(x^2-2)`. By the chain rule,

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

Quotient rule:

`u=(x^2+2)/(x^2-2)\implies u'=((x^2-2)(x^2+2)'-(x^2+2)(x^2-2)')/((x^2-2)^2)`

`u'=(2x(x^2-2)-2x(x^2+2))/((x^2-2)^2)=-(8x)/((x^2-2)^2)`

Putting everything together, we have

`y'=6\left((x^2+2)/(x^2-2)\right)^5\left(-(8x)/((x^2-2)^2)\right)`

`y'=-(48x(x^2+2)^5)/((x^2-2)^7)`