Question

What is the derivative of `(x-1)(x^2+2)^3`?

Answer 1

`(x^2+2)^2(7x^2-6x+2)`

Explanation:

`"differentiate using the "color(blue)"product/chain rules"`

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

`dy/dx=g(x)h'(x)+h(x)g'(x)larrcolor(blue)"product rule"`

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

`dy/dx=f'(g(x))xxg'(x)larrcolor(blue)"chain rule"`

`g(x)=x-1rArrg'(x)=1`

`h(x)=(x^2+2)^3rArrh'(x)=3(x^2+2)^2xx2x`

`color(white)(xxxxxxxxxxxxxxxxxx)=6x(x^2+2)^2`

`rArrd/dx((x-1)(x^2+2)^3)`

`=6x(x-1)(x^2+2)^2+(x^2+2)^3`

`=(x^2+2)^2[6x(x-1)+x^2+2]`

`=(x^2+2)^2(7x^2-6x+2)`