Home›Calculus 1›Second derivatives by implicit differentiation›Problem 2.2469

Second derivatives by implicit differentiation

Problem 2.2469 · hard

The curve \( \displaystyle 3 x^{2} + 2 y^{2} = 14 \) passes through \( \displaystyle (2, 1) \). Find \( \displaystyle y'' \) there.

\[ \left. 3 x^{2} + 2 y^{2} \right|_{\substack{ x=2\\ y=1 }} = 14 \]
The point is on the curve.✓ Proved
\[ - \frac{\frac{\partial}{\partial x} \left(3 x^{2} + 2 y^{2}\right)}{\frac{\partial}{\partial y} \left(3 x^{2} + 2 y^{2}\right)} = - \frac{3 x}{2 y} \]
y′ = −F_x/F_y.✓ Proved
\[ \frac{d}{d x} \left(- \frac{3 x}{2 Y{\left(x \right)}}\right) = \frac{3 x \frac{d}{d x} Y{\left(x \right)}}{2 Y^{2}{\left(x \right)}} - \frac{3}{2 Y{\left(x \right)}} \]
Differentiate y′ again, with y a function of x.✓ Proved
Replace dy/dx in that by y′ itself, and simplify.
Reviewed
\[ \left. \frac{3 \left(- 3 x^{2} - 2 y^{2}\right)}{4 y^{3}} \right|_{\substack{ x=2\\ y=1 }} = - \frac{21}{2} \]
At (2, 1).✓ Proved

Answer \( y'' = \frac{3 \left(- 3 x^{2} - 2 y^{2}\right)}{4 y^{3}},\quad y''(2) = - \frac{21}{2} \)

Lines: 4 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly applies implicit differentiation to find the second derivative and evaluates it at the given point. The steps are logically sound and the final result is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly applies implicit differentiation to find the second derivative and evaluates it at the given point. The steps are logically sound and the final result is correct.
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-mlx: fail (error) 2026-10-20 — The solution fails to substitute the original constraint 3x^2 + 2y^2 = 14 into the expression for y'', resulting in an incorrect general formula y'' = 3(-3x^2 - 2y^2)/(4y^3) instead of the correct y'' = -21/(4y^3). Although the final numerical answer is coincidentally correct because the constraint holds at the point, the derived formula is mathematically false.
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 final result is correct.
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.