Second derivatives by implicit differentiation
Problem 2.2254 · hard
The curve \( \displaystyle 3 x^{2} + 2 y^{2} = 5 \) passes through \( \displaystyle (1, 1) \). Find \( \displaystyle y'' \) there.
- \[ \left. 3 x^{2} + 2 y^{2} \right|_{\substack{ x=1\\ y=1 }} = 5 \]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
- \[ \left. \frac{3 \left(- 3 x^{2} - 2 y^{2}\right)}{4 y^{3}} \right|_{\substack{ x=1\\ y=1 }} = - \frac{15}{4} \]At (1, 1).✓ Proved
Answer \( y'' = \frac{3 \left(- 3 x^{2} - 2 y^{2}\right)}{4 y^{3}},\quad y''(1) = - \frac{15}{4} \)
✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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: passqwen3.6:27b-mlx: pass — The solution correctly applies implicit differentiation and the quotient rule. The substitution of the original equation constraint simplifies the expression correctly, leading to the right final value.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies implicit differentiation and the quotient rule. The substitution of the original equation constraint simplifies the expression correctly, leading to the right final value.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies implicit differentiation and the quotient rule. The substitution of the original equation constraint into the second derivative expression is a valid and efficient simplification step.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies implicit differentiation twice and substitutes the curve equation to simplify the second derivative before evaluating at the point.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.