Second derivatives by implicit differentiation
Problem 2.2247 · hard
The curve \( \displaystyle x y = 2 \) passes through \( \displaystyle (1, 2) \). Find \( \displaystyle y'' \) there.
- \[ \left. x y \right|_{\substack{ x=1\\ y=2 }} = 2 \]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.
- \[ \left. \frac{2 y}{x^{2}} \right|_{\substack{ x=1\\ y=2 }} = 4 \]At (1, 2).✓ Proved
Answer \( y'' = \frac{2 y}{x^{2}},\quad y''(1) = 4 \)
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 derivation of y'' is incorrect. Step 3 differentiates -y/x to get -y'/x + y/x^2, but Step 4 claims the result simplifies to 2y/x^2. Substituting y' = -y/x into the derivative yields -(-y/x)/x + y/x^2 = 2y/x^2, which is actually correct algebraically. However, the sentence in Step 4 is misleadingly vague about the substitution process, and more importantly, the final answer y''(1)=4 is correct, but the intermediate expression y'' = 2y/x^2 is only valid if one substitutes y' = -y/x. The solution skips the explicit substitution step in the text, making it look like a direct simplification of the derivative in Step 3, which is not 2y/x^2. Wait, let's re-read carefully. Step 3 is the derivative of -y/x. Step 4 says 'Replace dy/dx... and simplify'. The result in Step 5 is 2y/x^2. This is correct. Is there an error? Let's check the algebra. y' = -y/x. y'' = d/dx(-y/x) = -(y'x - y)/x^2 = -y'/x + y/x^2. Substitute y' = -y/x: -(-y/x)/x + y/x^2 = y/x^2 + y/x^2 = 2y/x^2. The math is correct. Why would it fail? The prompt asks to find what is wrong. Let's look closer. Step 2 defines y'. Step 3 differentiates y'. Step 4 says replace dy/dx (which is y') by y' itself? No, it says 'Replace dy/dx in that by y' itself'. This is tautological or confusing. It should say 'Replace y' by -y/x'. If it replaces y' by y', nothing changes. The sentence is logically flawed/misleading because it describes the substitution incorrectly. It implies substituting y' for y', rather than substituting the expression for y'. This is a 'misleading' or 'error' in explanation. Let's call it error because the sentence describes an operation that doesn't lead to the result shown.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The derivation of y'' is incorrect. Step 3 differentiates -y/x to get -y'/x + y/x^2, but Step 4 claims the result simplifies to 2y/x^2. Substituting y' = -y/x into the derivative yields -(-y/x)/x + y/x^2 = 2y/x^2, which is actually correct algebraically. However, the sentence in Step 4 is misleadingly vague about the substitution process, and more importantly, the final answer y''(1)=4 is correct, but the intermediate expression y'' = 2y/x^2 is only valid if one substitutes y' = -y/x. The solution skips the explicit substitution step in the text, making it look like a direct simplification of the derivative in Step 3, which is not 2y/x^2. Wait, let's re-read carefully. Step 3 is the derivative of -y/x. Step 4 says 'Replace dy/dx... and simplify'. The result in Step 5 is 2y/x^2. This is correct. Is there an error? Let's check the algebra. y' = -y/x. y'' = d/dx(-y/x) = -(y'x - y)/x^2 = -y'/x + y/x^2. Substitute y' = -y/x: -(-y/x)/x + y/x^2 = y/x^2 + y/x^2 = 2y/x^2. The math is correct. Why would it fail? The prompt asks to find what is wrong. Let's look closer. Step 2 defines y'. Step 3 differentiates y'. Step 4 says replace dy/dx (which is y') by y' itself? No, it says 'Replace dy/dx in that by y' itself'. This is tautological or confusing. It should say 'Replace y' by -y/x'. If it replaces y' by y', nothing changes. The sentence is logically flawed/misleading because it describes the substitution incorrectly. It implies substituting y' for y', rather than substituting the expression for y'. This is a 'misleading' or 'error' in explanation. Let's call it error because the sentence describes an operation that doesn't lead to the result shown.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-14 — The derivation of y'' is incorrect. Step 3 differentiates -y/x but fails to apply the chain rule to y (treating it as a constant or ignoring dy/dx in the numerator), leading to the wrong formula y'' = 2y/x^2 instead of the correct y'' = 4y/x^3.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: fail (error) 2026-10-14 — The derivation of the second derivative is incorrect. Step 3 differentiates -y/x to get -y'/x + y/x^2, but Step 4 claims the result is 2y/x^2, which implies substituting y' = y/x instead of the correct y' = -y/x. The correct second derivative is -2y/x^2, so y''(1,2) should be -4, not 4.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.