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: Understanding the Concept:
The notation \( \frac{\partial^2 z}{\partial x \partial y} \) represents a mixed partial derivative. This means we first differentiate \( z \) with respect to \( y \) (treating \( x \) as a constant), and then differentiate the result with respect to \( x \) (treating \( y \) as a constant).
Step 2: Key Formula or Approach:
1. First find \( \frac{\partial z}{\partial y} \).
2. Then find \( \frac{\partial}{\partial x} \left( \frac{\partial z}{\partial y} \right) \).
Step 3: Detailed Explanation:
Differentiate \( z = x^2 y^3 + e^y \sin x \) with respect to \( y \): \[ \frac{\partial z}{\partial y} = x^2(3y^2) + (\sin x)e^y = 3x^2 y^2 + e^y \sin x \] Now, differentiate this result with respect to \( x \): \[ \frac{\partial}{\partial x} (3x^2 y^2 + e^y \sin x) = (3y^2)(2x) + e^y (\cos x) \] \[ = 6xy^2 + e^y \cos x \]
Step 4: Final Answer:
The mixed partial derivative is \( 6xy^2 + e^y \cos x \).