Partial derivatives
Problem 10.3 · hard
For \( \displaystyle f(x, y) = \frac{x + y}{x - y + 3} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
- For f_x hold y constant; for f_y hold x constant.Reviewed
- \[ \frac{\partial}{\partial x} \frac{x + y}{x - y + 3} = \frac{3 - 2 y}{\left(x - y + 3\right)^{2}} \]f_x.✓ Proved
- \[ \frac{\partial}{\partial y} \frac{x + y}{x - y + 3} = \frac{2 x + 3}{\left(x - y + 3\right)^{2}} \]f_y.✓ Proved
- \[ \frac{\partial}{\partial y} \frac{3 - 2 y}{\left(x - y + 3\right)^{2}} = \frac{- 2 x - 2 y}{\left(x - y + 3\right)^{3}} \]f_xy: differentiate f_x with respect to y.✓ Proved
- \[ \frac{\partial}{\partial x} \frac{2 x + 3}{\left(x - y + 3\right)^{2}} = \frac{- 2 x - 2 y}{\left(x - y + 3\right)^{3}} \]Clairaut: f_yx is the same.✓ Proved
Answer \( f_x = \frac{3 - 2 y}{\left(x - y + 3\right)^{2}},\quad f_y = \frac{2 x + 3}{\left(x - y + 3\right)^{2}},\quad f_{xy} = \frac{2 \left(- x - y\right)}{\left(x - y + 3\right)^{3}} \)
✓ 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 | claude-sonnet-5-5 | claude-sonnet-5-5: A sentence stating the standard convention: hold y constant when differentiating in x, and x constant when differentiating in y. It is not an algebraic equivalence and it is correct. |
| 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: fail (error) — The partial derivatives are incorrectly computed: f_x and f_y are swapped, and the mixed derivative f_xy is wrong. The correct results are f_x=(2x+3)/(x-y+3)^2, f_y=(3-2y)/(x-y+3)^2, and f_xy=2(2x+3)/(x-y+3)^3.qwen3.6:27b-mlx: pass — The solution correctly applies the quotient rule for partial derivatives and verifies Clairaut's theorem. The setup and calculations are accurate.
Senior review claude-sonnet-5-5, 2026-09-30: pass — All derivatives check by hand: f_xy = (-2(x-y+3)+2(3-2y))/D^3 = -2(x+y)/D^3, and f_yx gives the same, so the Clairaut remark is valid. Line 1 is only a correct statement of the method.
gpt-oss:20b: dismiss — By the quotient rule f_x = ((x-y+3)-(x+y))/D^2 = (3-2y)/D^2 and f_y = ((x-y+3)+(x+y))/D^2 = (2x+3)/D^2, so the stated answer is right and the panel's 'corrected' values are wrong (they are swapped and the f_xy is false); SymPy proved lines 2-5.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-09-30 — All derivatives check by hand: f_xy = (-2(x-y+3)+2(3-2y))/D^3 = -2(x+y)/D^3, and f_yx gives the same, so the Clairaut remark is valid. Line 1 is only a correct statement of the method.qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the quotient rule for partial derivatives and verifies Clairaut's theorem. The setup and calculations are accurate.gpt-oss:20b: fail (error) 2026-09-28 — The partial derivatives are incorrectly computed: f_x and f_y are swapped, and the mixed derivative f_xy is wrong. The correct results are f_x=(2x+3)/(x-y+3)^2, f_y=(3-2y)/(x-y+3)^2, and f_xy=2(2x+3)/(x-y+3)^3.qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the partial derivatives and the mixed partial derivative. The application of Clairaut's theorem is valid since the function is rational and smooth away from the singularity.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the quotient rule for partial derivatives and verifies Clairaut's theorem. The setup and calculations are accurate.gpt-oss:20b: fail (error) 2026-09-26 — The partial derivatives are swapped: f_x should be (2x+3)/(x−y+3)^2 and f_y should be (3−2y)/(x−y+3)^2. Consequently the mixed partials computed from f_x and f_y are incorrect; the correct f_xy is 2(2x+3)/(x−y+3)^3, not 2(−x−y)/(x−y+3)^3.
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.