Final Answer:
The derivative dy/dx for the equation x³ - y + y² = 4 is d/dx(y) = (3x² - 1)/(2y - 1).
Step-by-step explanation:
To find the derivative dy/dx of the implicit function x³ - y + y² = 4, we employ implicit differentiation. Taking the derivative of each term with respect to x, we get 3x² - dy/dx + 2yy' = 0. Rearranging and solving for dy/dx, we have dy/dx = (3x² - 1)/(2y - 1).
This result is obtained by isolating the term involving y' and expressing it as the ratio of the derivative of x with respect to y over the derivative of y with respect to x. The derivative of x³ with respect to x is 3x², and the derivative of y² with respect to x is 2yy'. Solving for dy/dx provides the final expression.
In conclusion, the derivative dy/dx for the given equation is (3x² - 1)/(2y - 1), representing the rate of change of y with respect to x for the given implicit function