Step 1: Understanding the Concept:
This problem requires finding the second-order partial derivatives of a function and then evaluating their sum at a specific point. This involves applying the rules of partial differentiation twice.
Step 2: Key Formula or Approach:
1. Find the first partial derivatives, $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$.
2. Find the second partial derivatives, $\frac{\partial^2 u}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial u}{\partial x}\right)$ and $\frac{\partial^2 u}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial u}{\partial y}\right)$.
3. Add the second partial derivatives together.
4. Substitute the given point's coordinates into the resulting expression.
Step 3: Detailed Explanation:
The function is $u = e^{xy}$.
1. Find the first partial derivatives:
\[ \frac{\partial u}{\partial x} = \frac{\partial}{\partial x}(e^{xy}) = e^{xy} \cdot \frac{\partial}{\partial x}(xy) = y e^{xy} \]
\[ \frac{\partial u}{\partial y} = \frac{\partial}{\partial y}(e^{xy}) = e^{xy} \cdot \frac{\partial}{\partial y}(xy) = x e^{xy} \]
2. Find the second partial derivatives:
\[ \frac{\partial^2 u}{\partial x^2} = \frac{\partial}{\partial x}(y e^{xy}) = y \cdot \frac{\partial}{\partial x}(e^{xy}) = y \cdot (y e^{xy}) = y^2 e^{xy} \]
\[ \frac{\partial^2 u}{\partial y^2} = \frac{\partial}{\partial y}(x e^{xy}) = x \cdot \frac{\partial}{\partial y}(e^{xy}) = x \cdot (x e^{xy}) = x^2 e^{xy} \]
3. Add the second partial derivatives:
\[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = y^2 e^{xy} + x^2 e^{xy} = (x^2 + y^2)e^{xy} \]
4. Evaluate this expression at the point (1, 1):
\[ \left. (x^2 + y^2)e^{xy} \right|_{(1,1)} = (1^2 + 1^2)e^{(1)(1)} = (1+1)e^1 = 2e \]
Step 4: Final Answer:
The value of the expression at (1, 1) is 2e. Therefore, option (B) is correct.