Question

If \(z = x^2 - y^2\) then \( \frac{1}{x} \frac{\partial z}{\partial x} + \frac{1}{y} \frac{\partial z}{\partial y} = \)

• A. 1
• B. 2x + 2y
• C. 0
• D. 2x - 2y

Hint:

This problem type is a straightforward application of partial differentiation rules. Remember that when differentiating with respect to one variable, all other variables are treated as constants.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
Find ∂²u/∂x²+∂²u/∂y² at (1,1) for u=eˣʸ.

Step 2: Key Formula (Alternate):
Chain rule twice for each variable, then add and evaluate.

Step 3: Detailed Explanation:
∂u/∂x=yeˣʸ, ∂²u/∂x²=y²eˣʸ. Similarly ∂²u/∂y²=x²eˣʸ. Sum=(x²+y²)eˣʸ. At (1,1): (1+1)e¹=2e.

Step 4: Final Answer:
Value is 2e.