Final answer:
To find the derivative y' for the equation 2x³ - 3y³ = 6 using implicit differentiation, we differentiate both sides with respect to x and solve for y', yielding y' = 2x²/3y².
Step-by-step explanation:
To find y' by implicit differentiation for the equation 2x³ - 3y³ = 6, we will differentiate both sides of the equation with respect to x. Given that y is a function of x, we will apply the chain rule when differentiating terms involving y.
Step-by-step process:
- Differentiate 2x³ with respect to x: 6x².
- Differentiate -3y³ with respect to x: -9y²*y' (using the chain rule).
- Set the differential of the right side, 0, equal to the combined differentials of the left side: 6x² - 9y²*y' = 0.
- Solve for y': y' = rac{6x²}{9y²} = rac{2x²}{3y²}.
Therefore, the derivative y' with respect to x is ²/3y²}..