Answer
$$y''=(x^3-1)^3(272x^6-94x^3+2)$$
Work Step by Step
$$y=x^2(x^3-1)^5$$
- Find $y'$: $$y'=(x^2)'(x^3-1)^5+x^2\Big((x^3-1)^5\Big)'$$ $$y'=2x(x^3-1)^5+x^2\Big(5(x^3-1)^4(x^3-1)'\Big)$$ $$y'=2x(x^3-1)^5+5x^2(x^3-1)^4\times(3x^2)$$ $$y'=2x(x^3-1)^5+15x^4(x^3-1)^4$$ $$y'=(x^3-1)^4\Big(2x(x^3-1)+15x^4\Big)$$ $$y'=(x^3-1)^4(2x^4-2x+15x^4)$$ $$y'=(x^3-1)^4(17x^4-2x)$$
- Find $y''$: $$y''=\Big((x^3-1)^4\Big)'(17x^4-2x)+(x^3-1)^4(17x^4-2x)'$$ $$y''=\Big(4(x^3-1)^3(x^3-1)'\Big)(17x^4-2x)+(x^3-1)^4(68x^3-2)$$ $$y''=\Big(4(x^3-1)^3(3x^2)\Big)(17x^4-2x)+(x^3-1)^4(68x^3-2)$$ $$y''=12x^2(x^3-1)^3(17x^4-2x)+(x^3-1)^4(68x^3-2)$$ $$y''=(x^3-1)^3(204x^6-24x^3)+(x^3-1)^3(x^3-1)(68x^3-2)$$ $$y''=(x^3-1)^3(204x^6-24x^3)+(x^3-1)^3(68x^6-70x^3+2)$$ $$y''=(x^3-1)^3(272x^6-94x^3+2)$$
