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Partial derivatives

Problem 10.137 · hard

For \( \displaystyle f(x, y) = x^{2} \ln{\left(y + 1 \right)} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).

For f_x hold y constant; for f_y hold x constant.
Reviewed
\[ \frac{\partial}{\partial x} x^{2} \ln{\left(y + 1 \right)} = 2 x \ln{\left(y + 1 \right)} \]
f_x.✓ Proved
\[ \frac{\partial}{\partial y} x^{2} \ln{\left(y + 1 \right)} = \frac{x^{2}}{y + 1} \]
f_y.✓ Proved
\[ \frac{\partial}{\partial y} 2 x \ln{\left(y + 1 \right)} = \frac{2 x}{y + 1} \]
f_xy: differentiate f_x with respect to y.✓ Proved
\[ \frac{\partial}{\partial x} \frac{x^{2}}{y + 1} = \frac{2 x}{y + 1} \]
Clairaut: f_yx is the same.✓ Proved

Answer \( f_x = 2 x \ln{\left(y + 1 \right)},\quad f_y = \frac{x^{2}}{y + 1},\quad f_{xy} = \frac{2 x}{y + 1} \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
5	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	difference quotients of f at (0.6, 0.4) agree with f_x and f_y

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies partial differentiation rules and verifies the result using Clairaut's theorem. The setup and calculations are sound.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies partial differentiation rules and verifies the result using Clairaut's theorem. The setup and calculations are sound.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the partial derivatives and verifies Clairaut's theorem. The setup and steps are logically sound and mathematically correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies partial differentiation rules and verifies consistency via Clairaut's theorem. The steps are logically sound and algebraically correct.
gpt-oss:20b: pass 2026-09-26

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/partial_derivatives, checked 2026-09-28 with SymPy 1.14.0.