Final answer:
To find the second derivative (y'') using implicit differentiation, differentiate both sides of the equation with respect to x. Then, rearrange the equation to solve for y''.
Step-by-step explanation:
To find the second derivative (y'') using implicit differentiation, we start by differentiating both sides of the equation with respect to x. The derivative of x^2 is 2x, and the derivative of -3y^2 is -6yy'. Finally, the derivative of 3y'' is 3y'''. Therefore, our equation becomes 2x - 6yy' = 3y'''.
Next, we solve this equation for y''. We rearrange the terms and divide by 3, resulting in y''' = (2x - 6yy') / 3.
So, the second derivative y'' is given by (2x - 6yy') / 3.