Question

How do you differentiate `g(x) =(x^2 - 1) (x^2 - 2)^(3/2` using the product rule?

Answer 1

`g'(x)=x(x^2-2)^(1/2)(5x^2-7)`

Explanation:

`"given "g(x)=f(x)h(x)" then"`

`g'(x)=f(x)h'(x)+h(x)f'(x)larrcolor(blue)"product rule"`

`f(x)=x^2-1rArrf'(x)=2x`

`h(x)=(x^2-2)^(3/2)`

`"differentiate using the "color(blue)"chain rule"`

`rArrh'(x)=3/2(x^2-2)^(1/2)xx2x=3x(x^2-2)^(1/2)`

`rArrg'(x)=3x(x^2-1)(x^2-2)^(1/2)+2x(x^2-2)^(3/2)`

`color(white)(rArrg'(x))=x(x^2-2)^(1/2)[3(x^2-1)+2(x^2-2)]`

`color(white)(rArrg'(x))=x(x^2-2)^(1/2)(5x^2-7)`