Question

If \(z = 3x^2y - x \sin xy\), then:

• A. \(\frac{\partial^2 z}{\partial y \partial x} > \frac{\partial^2 z}{\partial x \partial y}\)
• B. \(\frac{\partial^2 z}{\partial y \partial x} < \frac{\partial^2 z}{\partial x \partial y}\)
• C. \(\frac{\partial^2 z}{\partial y \partial x} = \frac{\partial^2 z}{\partial x \partial y}\)
• D. \(\frac{\partial^2 z}{\partial y \partial x} = 0\)

Hint:

For any standard, well-behaved mathematical function composed of polynomials, exponentials, sines, and cosines, the mixed partial derivatives are always equal:
\[ \frac{\partial^2 z}{\partial y \partial x} = \frac{\partial^2 z}{\partial x \partial y} \]
You can select this option immediately without doing any differentiation.

Answer 1

The Correct Option is C