Home›Calculus 1›Second derivatives by implicit differentiation›Problem 2.2471

Second derivatives by implicit differentiation

Problem 2.2471 · hard

The curve \( \displaystyle x^{2} + 2 y^{2} = 9 \) passes through \( \displaystyle (-1, 2) \). Find \( \displaystyle y'' \) there.

\[ \left. x^{2} + 2 y^{2} \right|_{\substack{ x=-1\\ y=2 }} = 9 \]
The point is on the curve.✓ Proved
\[ - \frac{\frac{\partial}{\partial x} \left(x^{2} + 2 y^{2}\right)}{\frac{\partial}{\partial y} \left(x^{2} + 2 y^{2}\right)} = - \frac{x}{2 y} \]
y′ = −F_x/F_y.✓ Proved
\[ \frac{d}{d x} \left(- \frac{x}{2 Y{\left(x \right)}}\right) = \frac{x \frac{d}{d x} Y{\left(x \right)}}{2 Y^{2}{\left(x \right)}} - \frac{1}{2 Y{\left(x \right)}} \]
Differentiate y′ again, with y a function of x.✓ Proved
Replace dy/dx in that by y′ itself, and simplify.
\[ \left. \frac{- x^{2} - 2 y^{2}}{4 y^{3}} \right|_{\substack{ x=-1\\ y=2 }} = - \frac{9}{32} \]
At (-1, 2).✓ Proved

Answer \( y'' = \frac{- x^{2} - 2 y^{2}}{4 y^{3}},\quad y''(-1) = - \frac{9}{32} \)

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 solution fails to explicitly substitute the point (-1, 2) into the first derivative expression before differentiating, or to substitute the specific value of y' into the second derivative formula. While the final algebraic expression for y'' is correct, the step from line 3 to 5 skips the crucial substitution of y' = -x/(2y) and the simplification process, making the derivation opaque and potentially misleading about how the specific numerical answer is derived from the general formula.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-21 — The solution fails to explicitly substitute the point (-1, 2) into the first derivative expression before differentiating, or to substitute the specific value of y' into the second derivative formula. While the final algebraic expression for y'' is correct, the step from line 3 to 5 skips the crucial substitution of y' = -x/(2y) and the simplification process, making the derivation opaque and potentially misleading about how the specific numerical answer is derived from the general formula.
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly applies implicit differentiation twice and substitutes the given point values accurately.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly applies implicit differentiation to find the second derivative and evaluates it at the given point. The algebraic steps are verified, and the logic is sound.
gpt-oss:20b: pass 2026-10-20

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 2026-10-21 with SymPy 1.14.0.