Question

If f(x, y) \(=\) x\(^3\)y \(+\) xy\(^3\) \(+\) e\(^{xy^2}\), then f\(_y\)(1, 1) is

• A. 2 \(+\) 2e
• B. 2 \(+\) 4e
• C. 4 \(+\) 2e
• D. 4 \(+\) 4e

Hint:

When taking partial derivatives:
- Keep the non-differentiating variable strictly constant in your mind.
- Verify the chain rule component: \( \frac{d}{du}(e^u) = e^u \frac{du}{dy} \).
Double check simple arithmetic to avoid losing easy marks.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question asks to calculate the first-order partial derivative of a function \( f(x, y) \) with respect to \( y \) (denoted as \( f_y \)), evaluated at the point \( (1, 1) \).

Step 2: Key Formula or Approach:
The partial derivative with respect to \( y \), \( f_y(x, y) = \frac{\partial f}{\partial y} \), is computed by treating the variable \( x \) as a constant and differentiating the function with respect to \( y \).
We will use standard differentiation rules:
- Power rule: \( \frac{\partial}{\partial y}(y^n) = n y^{n-1} \)
- Chain rule: \( \frac{\partial}{\partial y}\left(e^{u(y)}\right) = e^{u(y)} \cdot \frac{\partial u}{\partial y} \)

Step 3: Detailed Explanation:

• Differentiate each term with respect to \( y \):
- The given function is:
\[ f(x, y) = x^3 y + x y^3 + e^{xy^2} \]
- Differentiating the first term \( x^3 y \):
\[ \frac{\partial}{\partial y}(x^3 y) = x^3 \cdot (1) = x^3 \]
- Differentiating the second term \( x y^3 \):
\[ \frac{\partial}{\partial y}(x y^3) = x \cdot (3y^2) = 3xy^2 \]
- Differentiating the third term \( e^{xy^2} \) using the chain rule:
\[ \frac{\partial}{\partial y}\left(e^{xy^2}\right) = e^{xy^2} \cdot \frac{\partial}{\partial y}(xy^2) = e^{xy^2} \cdot (2xy) = 2xy e^{xy^2} \]
- Combine all the differentiated terms:
\[ f_y(x, y) = x^3 + 3xy^2 + 2xy e^{xy^2} \]

• Evaluate at the point \( (x, y) = (1, 1) \):
- Substitute \( x = 1 \) and \( y = 1 \):
\[ f_y(1, 1) = (1)^3 + 3(1)(1)^2 + 2(1)(1) e^{(1)(1)^2} \]
\[ f_y(1, 1) = 1 + 3 + 2 e^1 \]
\[ f_y(1, 1) = 4 + 2e \]


Step 4: Final Answer:
The value of the partial derivative \( f_y(1, 1) \) is \( 4 + 2e \).
Therefore, the correct choice is option (C).