Question

Find the second derivative of 2x^3-3y^2=8​

Answer 1

`(d^2y)/(dx^2)=(x(2y-x^3))/(y^3)`

Explanation:

Find the first implicit derivative using implicit differentiation

`2x^3-3y^2=8\\\\6x^2-6y(dy)/(dx)=0\\ \\-6y(dy)/(dx)=-6x^2\\ \\(dy)/(dx)=(x^2)/(y)`

Use the substitution of dy/dx to find the second derivative (d²y/dx²)

`(d^2y)/(dx^2)=((y)(2x)-(x^2)((dy)/(dx)))/(y^2)\\ \\(d^2y)/(dx^2)=(2xy-(x^2)((x^2)/(y)))/(y^2)\\\\(d^2y)/(dx^2)=(2xy-(x^4)/(y))/(y^2)\\\\(d^2y)/(dx^2)=(x(2y-x^3))/(y^3)`