Question Bank • QB_SUB_EM_0032 NAT • 2 Marks

Section 1: Engineering Mathematics

Calculus

Subtopic: Partial derivatives

Question Statement:

For the multivariable function $f(x, y) = x^2 y + y^3 x$, the value of the mixed second partial derivative $\frac{\partial^2 f}{\partial x \partial y}$ at $(x, y) = (1, 1)$ is:

Answer Key:

5.0

Step-by-Step Solution & Explanation

First partial with respect to $y$: $\frac{\partial f}{\partial y} = x^2 + 3 y^2 x$ Differentiating with respect to $x$: $\frac{\partial^2 f}{\partial x \partial y} = 2x + 3 y^2$ At $(1, 1)$: $2(1) + 3(1)^2 = 2 + 3 = 5.00$
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