Home›Calculus 1›Second derivatives by implicit differentiation›Problem 2.2255

Second derivatives by implicit differentiation

Problem 2.2255 · hard

The curve \( \displaystyle x^{3} + y^{3} = 9 \) passes through \( \displaystyle (1, 2) \). Find \( \displaystyle y'' \) there.

\[ \left. x^{3} + y^{3} \right|_{\substack{ x=1\\ y=2 }} = 9 \]
The point is on the curve.✓ Proved
\[ - \frac{\frac{\partial}{\partial x} \left(x^{3} + y^{3}\right)}{\frac{\partial}{\partial y} \left(x^{3} + y^{3}\right)} = - \frac{x^{2}}{y^{2}} \]
y′ = −F_x/F_y.✓ Proved
\[ \frac{d}{d x} \left(- \frac{x^{2}}{Y^{2}{\left(x \right)}}\right) = \frac{2 x^{2} \frac{d}{d x} Y{\left(x \right)}}{Y^{3}{\left(x \right)}} - \frac{2 x}{Y^{2}{\left(x \right)}} \]
Differentiate y′ again, with y a function of x.✓ Proved
Replace dy/dx in that by y′ itself, and simplify.
\[ \left. \frac{2 x \left(- x^{3} - y^{3}\right)}{y^{5}} \right|_{\substack{ x=1\\ y=2 }} = - \frac{9}{16} \]
At (1, 2).✓ Proved

Answer \( y'' = \frac{2 x \left(- x^{3} - y^{3}\right)}{y^{5}},\quad y''(1) = - \frac{9}{16} \)

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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The derivative calculation in line 3 is incorrect. The derivative of -x^2 y^{-2} with respect to x is -2x y^{-2} + 2x^2 y^{-3} y', not 2x^2 y' y^{-3

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-17 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The derivative calculation in line 3 is incorrect. The derivative of -x^2 y^{-2} with respect to x is -2x y^{-2} + 2x^2 y^{-3} y', not 2x^2 y' y^{-3
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The differentiation in step 3 is incorrect; the derivative of -x^2/y^2 is (-2xy^2 - x^2(2y)y')/y^4, not the expression shown. Additionally, the final formula in step 5 is missing a factor of 3 from the original implicit differentiation (y' = -3x^2/3y^2), leading to an incorrect numerical result.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The differentiation in step 3 is incorrect; the derivative of -x^2/y^2 is (-2xy^2 + 2x^2y y')/y^4, not the expression shown. Consequently, the final formula for y'' is wrong.
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.