Second derivatives by implicit differentiation
Problem 2.2256 · hard
The curve \( \displaystyle x^{3} + y^{3} = 2 \) passes through \( \displaystyle (1, 1) \). Find \( \displaystyle y'' \) there.
- \[ \left. x^{3} + y^{3} \right|_{\substack{ x=1\\ y=1 }} = 2 \]The point is on the curve.✓ Proved
- \[ - \frac{\frac{\partial}{\partial x} \left(x^{3} + y^{3}\right)}{\frac{\partial}{\partial y} \left(x^{3} + y^{3}\right)} = - \frac{x^{2}}{y^{2}} \]y′ = −F_x/F_y.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{x^{2}}{Y^{2}{\left(x \right)}}\right) = \frac{2 x^{2} \frac{d}{d x} Y{\left(x \right)}}{Y^{3}{\left(x \right)}} - \frac{2 x}{Y^{2}{\left(x \right)}} \]Differentiate y′ again, with y a function of x.✓ Proved
- Replace dy/dx in that by y′ itself, and simplify.Reviewed
- \[ \left. \frac{2 x \left(- x^{3} - y^{3}\right)}{y^{5}} \right|_{\substack{ x=1\\ y=1 }} = -4 \]At (1, 1).✓ Proved
Answer \( y'' = \frac{2 x \left(- x^{3} - y^{3}\right)}{y^{5}},\quad y''(1) = -4 \)
Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 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 | y solved numerically from the curve at x₀ and x₀ ± 10⁻⁶, then a second difference |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies implicit differentiation twice and evaluates the result at the given point. The intermediate steps and final answer are correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies implicit differentiation twice and evaluates the result at the given point. The intermediate steps and final answer are correct.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-14 — The differentiation in step 3 is algebraically incorrect; the derivative of -x^2/y^2 with respect to x is (2xy^2 + 2x^2y y')/y^4, not the expression shown. Consequently, the final formula for y'' is wrong.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: fail (error) 2026-10-14 — The differentiation in step 3 is incorrect; the derivative of -x^2/y^2 with respect to x is (2xy^2 - 2x^2y y')/y^4, not the expression shown. Consequently, the final formula for y'' is wrong.gpt-oss:20b: fail (error) 2026-10-14 — The general formula for y'' derived in step 4 is incorrect. The correct second derivative is y'' = -2x/y^2 - 2x^4/y^5, which only equals the proposed expression 2x(-x^3 - y^3)/y^5 at the specific point (1,1).
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/implicit_second_derivative, checked 2026-10-17 with SymPy 1.14.0.