Home›Calculus 1›Second derivatives by implicit differentiation›Problem 2.2465

Second derivatives by implicit differentiation

Problem 2.2465 · hard

The curve $ \displaystyle 2 x^{2} + 3 y^{2} = 11 $ passes through $ \displaystyle (2, 1) $. Find $ \displaystyle y'' $ there.

\[ \left. 2 x^{2} + 3 y^{2} \right|_{\substack{ x=2\\ y=1 }} = 11 \]
The point is on the curve.✓ Proved
\[ - \frac{\frac{\partial}{\partial x} \left(2 x^{2} + 3 y^{2}\right)}{\frac{\partial}{\partial y} \left(2 x^{2} + 3 y^{2}\right)} = - \frac{2 x}{3 y} \]
y′ = −F_x/F_y.✓ Proved
\[ \frac{d}{d x} \left(- \frac{2 x}{3 Y{\left(x \right)}}\right) = \frac{2 x \frac{d}{d x} Y{\left(x \right)}}{3 Y^{2}{\left(x \right)}} - \frac{2}{3 Y{\left(x \right)}} \]
Differentiate y′ again, with y a function of x.✓ Proved
Replace dy/dx in that by y′ itself, and simplify.
\[ \left. \frac{2 \left(- 2 x^{2} - 3 y^{2}\right)}{9 y^{3}} \right|_{\substack{ x=2\\ y=1 }} = - \frac{22}{9} \]
At (2, 1).✓ Proved

Answer $ y'' = \frac{2 \left(- 2 x^{2} - 3 y^{2}\right)}{9 y^{3}},\quad y''(2) = - \frac{22}{9} $

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 substitute the original constraint 2x^2 + 3y^2 = 11 into the expression for y'' before evaluating at the point. While the final numerical answer is correct, the symbolic formula provided in the stated answer (and implied by the solution flow) is incorrect because it leaves the term (-2x^2 - 3y^2) unsimplified. A student following this would learn that y'' = -2(2x^2+3y^2)/(9y^3) is the general formula, which is false; the correct general formula is y'' = -22/(9y^3).

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-21 — The solution fails to substitute the original constraint 2x^2 + 3y^2 = 11 into the expression for y'' before evaluating at the point. While the final numerical answer is correct, the symbolic formula provided in the stated answer (and implied by the solution flow) is incorrect because it leaves the term (-2x^2 - 3y^2) unsimplified. A student following this would learn that y'' = -2(2x^2+3y^2)/(9y^3) is the general formula, which is false; the correct general formula is y'' = -22/(9y^3).
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-mlx: pass 2026-10-20
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly applies implicit differentiation twice. The algebraic simplification in step 4 is valid because the original equation $2x^2 + 3y^2 = 11$ allows substituting $-11$ for $-2x^2 - 3y^2$, leading to the correct final value.
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.