Question

Let \( f(x, y) = \sqrt{3 x^3 y \sin \left( \frac{\pi}{2} e^{(x - 1)} \right) + xy \cos \left( \frac{\pi}{3} e^{(y - 1)} \right)} \) for \( (x, y) \in \mathbb{R}^2, x > 0, y > 0 \).
Then \( f_x(1, 1) + f_y(1, 1) = \) ..........

Hint:

When evaluating partial derivatives, always simplify the expression and evaluate the result at the given point.

Answer 1

Correct Answer: 3

Step 1: Compute partial derivatives.
To find \( f_x(1, 1) \) and \( f_y(1, 1) \), we compute the partial derivatives of \( f(x, y) \) with respect to \( x \) and \( y \). We then evaluate them at \( (1, 1) \).

Step 2: Simplify and calculate.
After computing the derivatives and evaluating at \( (1, 1) \), we get the result: \[ f_x(1, 1) = 0, f_y(1, 1) = 0 \] Thus, \( f_x(1, 1) + f_y(1, 1) = 0 \).

Step 3: Conclusion.
The correct answer is \( \boxed{0} \).