A scalar field is defined by $f(x,y) = 3x^2 y - 2y^3 + 5x$. What is the numerical value of the mixed second-order partial derivative $\frac{\partial^2 f}{\partial x \partial y}$ evaluated at the coordinates $(x,y) = (2, 3)$?

Answer Key:

12

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: First Partial w.r.t $y$:** $\frac{\partial}{\partial y}(3x^2 y - 2y^3 + 5x) = 3x^2 - 6y^2$. 2. **Stage 2: Second Partial w.r.t $x$:** $\frac{\partial}{\partial x}(3x^2 - 6y^2) = 6x$. 3. **Stage 3: Evaluation at $(2, 3)$:** $6(2) = 12$. 4. **Stage 4: Result:** 12.

← Prev Question Next Question →