Question

If \( Z = 2x^2y^2 + 3xy^2 + y^3 \). The partial derivative of Z with respect to x is :

• A. \( 4xy^2 + 3y^2 \)
• B. \( 4x^2y + 6xy + 3y^2 \)
• C. \( 4y^2 \)
• D. \( 4x^2 + 6x + 6y \)

Hint:

Treat y as a constant and differentiate each term with respect to x.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:

• For the partial derivative with respect to \( x \) we treat \( y \) as a constant.

Step 2: Detailed Explanation:

• \( \frac{\partial}{\partial x}(2x^2y^2) = 4xy^2 \)

• \( \frac{\partial}{\partial x}(3xy^2) = 3y^2 \)

• \( \frac{\partial}{\partial x}(y^3) = 0 \)

• Adding, \( \frac{\partial Z}{\partial x} = 4xy^2 + 3y^2 \).

Step 3: Final Answer:
The correct option is (A) \( 4xy^2 + 3y^2 \).