Partial derivatives
Problem 10.123 · hard
For \( \displaystyle f(x, y) = 5 x^{2} y - 4 x y^{3} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
- For f_x hold y constant; for f_y hold x constant.Reviewed
- \[ \frac{\partial}{\partial x} \left(5 x^{2} y - 4 x y^{3}\right) = 10 x y - 4 y^{3} \]f_x.✓ Proved
- \[ \frac{\partial}{\partial y} \left(5 x^{2} y - 4 x y^{3}\right) = x \left(5 x - 12 y^{2}\right) \]f_y.✓ Proved
- \[ \frac{\partial}{\partial y} \left(10 x y - 4 y^{3}\right) = 10 x - 12 y^{2} \]f_xy: differentiate f_x with respect to y.✓ Proved
- \[ \frac{\partial}{\partial x} x \left(5 x - 12 y^{2}\right) = 10 x - 12 y^{2} \]Clairaut: f_yx is the same.✓ Proved
Answer \( f_x = 10 x y - 4 y^{3},\quad f_y = x \left(5 x - 12 y^{2}\right),\quad f_{xy} = 10 x - 12 y^{2} \)
✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the partial derivatives and the mixed partial derivative. The application of Clairaut's theorem is valid as the function is a polynomial, ensuring continuous second partial derivatives.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the partial derivatives and the mixed partial derivative. The application of Clairaut's theorem is valid as the function is a polynomial, ensuring continuous second partial derivatives.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the partial derivatives and verifies the equality of mixed partials using Clairaut's theorem. The setup and calculations are sound.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly computes the partial derivatives and the mixed partial derivative. The application of Clairaut's theorem is valid as the function is a polynomial, and the final results match the stated answer.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.