Question

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

• A. \(6xy^2+e^y\cos x\)
• B. \(3x^2y^2+e^y\sin x\)
• C. \(3x^2y^2+e^y\cos x\)
• D. \(6xy^2+e^y\sin x\)

Hint:

In partial derivatives, treat the other variable as constant while differentiating.

Answer 1

The Correct Option is A

Concept: For mixed partial derivative: \[ \frac{\partial^2 z}{\partial x\partial y} \] we first differentiate with respect to \(x\), then differentiate the result with respect to \(y\).

Step 1: Given: \[ z=x^2y^3+e^y\sin x \]

Step 2: Differentiate with respect to \(x\). \[ \frac{\partial z}{\partial x} = 2xy^3+e^y\cos x \]

Step 3: Now differentiate with respect to \(y\). \[ \frac{\partial}{\partial y}\left(2xy^3+e^y\cos x\right) \] \[ =6xy^2+e^y\cos x \] Therefore, \[ \boxed{6xy^2+e^y\cos x} \]