Home›Calculus 1›Second derivatives by implicit differentiation›Problem 2.2250

Second derivatives by implicit differentiation

Problem 2.2250 · hard

The curve \( \displaystyle x^{2} + 4 y^{2} = 5 \) passes through \( \displaystyle (1, -1) \). Find \( \displaystyle y'' \) there.

\[ \left. x^{2} + 4 y^{2} \right|_{\substack{ x=1\\ y=-1 }} = 5 \]
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.
\[ \left. \frac{- x^{2} - 4 y^{2}}{16 y^{3}} \right|_{\substack{ x=1\\ y=-1 }} = \frac{5}{16} \]
At (1, -1).✓ Proved

Answer \( y'' = \frac{- x^{2} - 4 y^{2}}{16 y^{3}},\quad y''(1) = \frac{5}{16} \)

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: pass
qwen3.6:27b-mlx: fail (error) — The final answer y''(1) = 5/16 is incorrect; the correct value is -5/16. The error stems from substituting y = -1 into the denominator 16y^3, where (-1)^3 = -1, resulting in a negative denominator, but the solution claims a positive result.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The final answer y''(1) = 5/16 is incorrect; the correct value is -5/16. The error stems from substituting y = -1 into the denominator 16y^3, where (-1)^3 = -1, resulting in a negative denominator, but the solution claims a positive result.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The final evaluation is incorrect: substituting x=1 and y=-1 into the derived formula yields -5/16, not 5/16. The sign error in the final answer makes the solution wrong.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The final answer y''(1) = 5/16 is incorrect; the correct value is -5/16. The error stems from substituting y = -1 into y^3 in the denominator, which yields -1, but the solution likely ignored the sign or made an arithmetic error in the final substitution step.
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.