Partial derivatives
Problem 10.140 · hard
For \( \displaystyle f(x, y) = 5 x^{2} y + 3 x y^{3} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
- For f_x hold y constant; for f_y hold x constant.Reviewed
- \[ \frac{\partial}{\partial x} \left(5 x^{2} y + 3 x y^{3}\right) = y \left(10 x + 3 y^{2}\right) \]f_x.✓ Proved
- \[ \frac{\partial}{\partial y} \left(5 x^{2} y + 3 x y^{3}\right) = x \left(5 x + 9 y^{2}\right) \]f_y.✓ Proved
- \[ \frac{\partial}{\partial y} y \left(10 x + 3 y^{2}\right) = 10 x + 9 y^{2} \]f_xy: differentiate f_x with respect to y.✓ Proved
- \[ \frac{\partial}{\partial x} x \left(5 x + 9 y^{2}\right) = 10 x + 9 y^{2} \]Clairaut: f_yx is the same.✓ Proved
Answer \( f_x = y \left(10 x + 3 y^{2}\right),\quad f_y = x \left(5 x + 9 y^{2}\right),\quad f_{xy} = 10 x + 9 y^{2} \)
✓ 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 applies the rules for partial differentiation and verifies the equality of mixed partials via Clairaut's theorem. The setup and calculations are sound.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the rules for partial differentiation and verifies the equality of mixed partials via 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 reference to Clairaut's theorem is appropriate given the smoothness of the polynomial function.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the rules for partial differentiation and verifies the mixed partials using Clairaut's theorem. The setup and calculations are sound.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 2026-09-27 with SymPy 1.14.0.