Second derivatives by implicit differentiation
Problem 2.2458 · hard
The curve \( \displaystyle x^{2} + 4 y^{2} = 8 \) passes through \( \displaystyle (2, 1) \). Find \( \displaystyle y'' \) there.
- \[ \left. x^{2} + 4 y^{2} \right|_{\substack{ x=2\\ y=1 }} = 8 \]The point is on the curve.✓ Proved
- \[ - \frac{\frac{\partial}{\partial x} \left(x^{2} + 4 y^{2}\right)}{\frac{\partial}{\partial y} \left(x^{2} + 4 y^{2}\right)} = - \frac{x}{4 y} \]y′ = −F_x/F_y.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{x}{4 Y{\left(x \right)}}\right) = \frac{x \frac{d}{d x} Y{\left(x \right)}}{4 Y^{2}{\left(x \right)}} - \frac{1}{4 Y{\left(x \right)}} \]Differentiate y′ again, with y a function of x.✓ Proved
- Replace dy/dx in that by y′ itself, and simplify.
- \[ \left. \frac{- x^{2} - 4 y^{2}}{16 y^{3}} \right|_{\substack{ x=2\\ y=1 }} = - \frac{1}{2} \]At (2, 1).✓ Proved
Answer \( y'' = \frac{- x^{2} - 4 y^{2}}{16 y^{3}},\quad y''(2) = - \frac{1}{2} \)
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: passqwen3.6:27b-mlx: fail (error) — Step 3 contains an algebraic error in the differentiation of y'. The derivative of -x/(4y) is (-4y - x(-4y'))/(16y^2) = (-y + xy')/(4y^2), but the solution incorrectly computes it as (xy')/(4y^2) - 1/(4y). Although the final answer is correct due to a coincidental cancellation at the specific point, the general formula derived in step 3 is wrong.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — Step 3 contains an algebraic error in the differentiation of y'. The derivative of -x/(4y) is (-4y - x(-4y'))/(16y^2) = (-y + xy')/(4y^2), but the solution incorrectly computes it as (xy')/(4y^2) - 1/(4y). Although the final answer is correct due to a coincidental cancellation at the specific point, the general formula derived in step 3 is wrong.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies implicit differentiation twice and substitutes the point values accurately.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies implicit differentiation to find the second derivative and evaluates it at the given point. The algebraic steps are sound and the final result is correct.gpt-oss:20b: pass 2026-10-17
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 2026-10-19 with SymPy 1.14.0.