Question

If $z = x^{2}y^{3} + e^{y}\sin x$, then $\frac{\partial^{2}z}{\partial x\partial y} =$

• A. $6xy^{2} + e^{y}\cos x$
• B. $3x^{2}y^{2} + e^{y}\sin x$
• C. $3x^{2}y^{2} + e^{y}\cos x$
• D. $6xy^{2} + e^{y}\sin x$

Hint:

For most smooth functions, $\frac{\partial^2 z}{\partial x \partial y} = \frac{\partial^2 z}{\partial y \partial x}$. You can pick whichever order is easier to calculate!

Answer 1

The Correct Option is A

Step 1: Concept
This is a mixed partial derivative. We differentiate first with respect to $y$, then with respect to $x$.

Step 2: Meaning
Treat $x$ as a constant when differentiating with respect to $y$, and vice-versa.

Step 3: Analysis
First, $\frac{\partial z}{\partial y} = 3x^2y^2 + e^y \sin x$. Now, differentiate this result with respect to $x$: $\frac{\partial}{\partial x}(3x^2y^2 + e^y \sin x) = 6xy^2 + e^y \cos x$.

Step 4: Conclusion
The mixed partial derivative matches Option (A).
Final Answer: (A)