Question

If \(u = e^{xy}\), then the value of \(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\) at (1, 1) is

• A. e
• B. 2e
• C. 1
• D. 0

Hint:

When performing partial differentiation, be methodical. Write down each step clearly to avoid confusion, especially when calculating second-order derivatives. Pay close attention to which variable you are treating as a constant in each step.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to find the sum of the second partial derivatives of the function \(u=e^{xy}\) with respect to \(x\) and \(y\), and then evaluate this sum at the point (1, 1).

Step 2: Key Formula or Approach:
1. Find the first partial derivative \(\frac{\partial u}{\partial x}\).
2. Find the second partial derivative \(\frac{\partial^2 u}{\partial x^2}\) by differentiating \(\frac{\partial u}{\partial x}\) with respect to \(x\).
3. Find the first partial derivative \(\frac{\partial u}{\partial y}\).
4. Find the second partial derivative \(\frac{\partial^2 u}{\partial y^2}\) by differentiating \(\frac{\partial u}{\partial y}\) with respect to \(y\).
5. Add the two second derivatives and substitute \(x=1, y=1\).

Step 3: Detailed Explanation:
The function is \(u = e^{xy}\).

Derivatives with respect to x:
First partial derivative w.r.t. x (using chain rule, treat y as constant):
\[ \frac{\partial u}{\partial x} = e^{xy} \cdot \frac{\partial}{\partial x}(xy) = y e^{xy} \]
Second partial derivative w.r.t. x (treat y as constant):
\[ \frac{\partial^2 u}{\partial x^2} = \frac{\partial}{\partial x}(y e^{xy}) = y \cdot \frac{\partial}{\partial x}(e^{xy}) = y \cdot (y e^{xy}) = y^2 e^{xy} \]

Derivatives with respect to y:
First partial derivative w.r.t. y (using chain rule, treat x as constant):
\[ \frac{\partial u}{\partial y} = e^{xy} \cdot \frac{\partial}{\partial y}(xy) = x e^{xy} \]
Second partial derivative w.r.t. y (treat x as constant):
\[ \frac{\partial^2 u}{\partial y^2} = \frac{\partial}{\partial y}(x e^{xy}) = x \cdot \frac{\partial}{\partial y}(e^{xy}) = x \cdot (x e^{xy}) = x^2 e^{xy} \]

Sum of second derivatives:
\[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = y^2 e^{xy} + x^2 e^{xy} = (x^2 + y^2)e^{xy} \]
Now, evaluate this expression at the point (1, 1):
\[ \left. \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right|_{(1,1)} = (1^2 + 1^2)e^{(1)(1)} = (1+1)e^1 = 2e \]

Step 4: Final Answer:
The value of the expression at (1, 1) is 2e.