Question:
Match the partial derivative of function \(Z=x^2+xy^2+y\).
List-I List-II A. \(Z_x\) I. \(2y\) B. \(Z_y\) II. \(2xy+1\) C. \(Z_{xy}\) III. \(2x+y^2\) D. \(Z_{yx}\) IV. \(2y\)
Match the partial derivative of function \(Z=x^2+xy^2+y\).
| List-I | List-II |
|---|---|
| A. \(Z_x\) | I. \(2y\) |
| B. \(Z_y\) | II. \(2xy+1\) |
| C. \(Z_{xy}\) | III. \(2x+y^2\) |
| D. \(Z_{yx}\) | IV. \(2y\) |
Show Hint
For well-behaved functions, mixed partial derivatives are equal: \(Z_{xy}=Z_{yx}\).
Updated On: May 22, 2026
- A-I, B-III, C-II, D-IV
- A-III, B-II, C-I, D-IV
- A-II, B-III, C-I, D-IV
- A-I, B-II, C-III, D-IV
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The Correct Option is B
Solution and Explanation
Concept: Partial derivative means differentiating a function with respect to one variable while treating the other variable as constant.
Step 1: Find \(Z_x\).
\[ Z=x^2+xy^2+y \] Differentiate with respect to \(x\): \[ Z_x=2x+y^2 \] So, \[ A \rightarrow III \]
Step 2: Find \(Z_y\).
Differentiate with respect to \(y\): \[ Z_y=2xy+1 \] So, \[ B \rightarrow II \]
Step 3: Find \(Z_{xy}\).
Differentiate \(Z_x=2x+y^2\) with respect to \(y\): \[ Z_{xy}=2y \] So, \[ C \rightarrow I \]
Step 4: Find \(Z_{yx}\).
Differentiate \(Z_y=2xy+1\) with respect to \(x\): \[ Z_{yx}=2y \] So, \[ D \rightarrow IV \] Therefore, the correct matching is: \[ A-III,\ B-II,\ C-I,\ D-IV \]
Step 1: Find \(Z_x\).
\[ Z=x^2+xy^2+y \] Differentiate with respect to \(x\): \[ Z_x=2x+y^2 \] So, \[ A \rightarrow III \]
Step 2: Find \(Z_y\).
Differentiate with respect to \(y\): \[ Z_y=2xy+1 \] So, \[ B \rightarrow II \]
Step 3: Find \(Z_{xy}\).
Differentiate \(Z_x=2x+y^2\) with respect to \(y\): \[ Z_{xy}=2y \] So, \[ C \rightarrow I \]
Step 4: Find \(Z_{yx}\).
Differentiate \(Z_y=2xy+1\) with respect to \(x\): \[ Z_{yx}=2y \] So, \[ D \rightarrow IV \] Therefore, the correct matching is: \[ A-III,\ B-II,\ C-I,\ D-IV \]
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