Second derivatives by implicit differentiation
Problem 2.2248 · hard
The curve \( \displaystyle x y = 6 \) passes through \( \displaystyle (2, 3) \). Find \( \displaystyle y'' \) there.
- \[ \left. x y \right|_{\substack{ x=2\\ y=3 }} = 6 \]The point is on the curve.✓ Proved
- \[ - \frac{\frac{\partial}{\partial x} x y}{\frac{\partial}{\partial y} x y} = - \frac{y}{x} \]y′ = −F_x/F_y.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{Y{\left(x \right)}}{x}\right) = - \frac{\frac{d}{d x} Y{\left(x \right)}}{x} + \frac{Y{\left(x \right)}}{x^{2}} \]Differentiate y′ again, with y a function of x.✓ Proved
- Replace dy/dx in that by y′ itself, and simplify.
- \[ \left. \frac{2 y}{x^{2}} \right|_{\substack{ x=2\\ y=3 }} = \frac{3}{2} \]At (2, 3).✓ Proved
Answer \( y'' = \frac{2 y}{x^{2}},\quad y''(2) = \frac{3}{2} \)
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: passqwen3.6:27b-mlx: fail (error) — The solution claims y'' = 2y/x^2, but the correct second derivative is y'' = 2y/x^3. The error arises in step 3/4 where the quotient rule is applied incorrectly or the substitution is mishandled, leading to a missing factor of 1/x.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution claims y'' = 2y/x^2, but the correct second derivative is y'' = 2y/x^3. The error arises in step 3/4 where the quotient rule is applied incorrectly or the substitution is mishandled, leading to a missing factor of 1/x.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution claims y'' = 2y/x^2, but the correct second derivative is 2y/x^3. The differentiation in step 3 is algebraically correct for the expression -y/x, but step 4 fails to substitute y' = -y/x correctly or simplifies incorrectly, leading to a missing factor of 1/x.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution claims y'' = 2y/x^2, but the correct second derivative is y'' = 2y/x^3. The differentiation in step 3 is algebraically correct for the expression -y/x, but step 4 fails to substitute y' = -y/x correctly into the result, or rather, the final simplified expression in step 5 is mathematically incorrect for the second derivative of xy=6.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 2026-10-17 with SymPy 1.14.0.