Second derivatives by implicit differentiation
Problem 2.2323 · hard
The curve \( \displaystyle 3 x^{2} + 3 y^{2} = 6 \) passes through \( \displaystyle (1, 1) \). Find \( \displaystyle y'' \) there.
- \[ \left. 3 x^{2} + 3 y^{2} \right|_{\substack{ x=1\\ y=1 }} = 6 \]The point is on the curve.✓ Proved
- \[ - \frac{\frac{\partial}{\partial x} \left(3 x^{2} + 3 y^{2}\right)}{\frac{\partial}{\partial y} \left(3 x^{2} + 3 y^{2}\right)} = - \frac{x}{y} \]y′ = −F_x/F_y.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{x}{Y{\left(x \right)}}\right) = \frac{x \frac{d}{d x} Y{\left(x \right)}}{Y^{2}{\left(x \right)}} - \frac{1}{Y{\left(x \right)}} \]Differentiate y′ again, with y a function of x.✓ Proved
- Replace dy/dx in that by y′ itself, and simplify.
- \[ \left. \frac{- x^{2} - y^{2}}{y^{3}} \right|_{\substack{ x=1\\ y=1 }} = -2 \]At (1, 1).✓ Proved
Answer \( y'' = \frac{- x^{2} - y^{2}}{y^{3}},\quad y''(1) = -2 \)
Lines: 4 proved, 1 not checked. 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 | Not checked | — | 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: fail (error) — The computed second derivative is incorrect. Substituting y′=−x/y into the derivative of y′ gives y″=(x^2−y^2)/y^3, not (−x^2−y^2)/y^3. Thus the solution’s conclusion is wrong.qwen3.6:27b-mlx: pass — The solution correctly applies implicit differentiation and the quotient rule to find the second derivative, and the final evaluation is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies implicit differentiation and the quotient rule to find the second derivative, and the final evaluation is correct.gpt-oss:20b: fail (error) 2026-10-18 — The computed second derivative is incorrect. Substituting y′=−x/y into the derivative of y′ gives y″=(x^2−y^2)/y^3, not (−x^2−y^2)/y^3. Thus the solution’s conclusion is wrong.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies implicit differentiation and the quotient rule to find the second derivative, and the final evaluation is correct.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies implicit differentiation to find the first and second derivatives, and the final evaluation is correct.gpt-oss:20b: pass 2026-10-14
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 2026-10-18 with SymPy 1.14.0.