Final answer:
The correct answer is option c. The derivative y' is found by applying the product rule to x²xy and the chain rule to y², leading to the simplified solution y' = (3 - x²y - y²)/xy, which corresponds to option c.
Step-by-step explanation:
To find the derivative y' by implicit differentiation of the equation x²xy + y² = 3, we first differentiate both sides of the equation with respect to x. Remember that y is a function of x, so we need to apply the product rule to the term x²xy and the chain rule to y².
Differentiating x²xy using the product rule gives us 2x·xy + x²·y' because the derivative of x² is 2x and the derivative of xy with respect to x is y + x·y' (applying the product rule again).
Next, the derivative of y² is 2y·y', as we use the chain rule (outer function y², with inner function y).
Setting the derivative of the entire left side equal to the derivative of the right side, we have: 2x·xy + x²·y' + 2y·y' = 0.
We now solve for y', which gives us: y' = (-2x·xy - 2yy)/(x² + 2y). Simplifying gives us: y' = (3 - x²y - y²)/(xy).
Therefore, the correct answer to the question is option c) y' = (3 - x²y - y²)/xy.