Home›Calculus 1›Second derivatives by implicit differentiation›Problem 2.2330

Second derivatives by implicit differentiation

Problem 2.2330 · hard

The curve \( \displaystyle 4 x^{2} + 2 y^{2} = 6 \) passes through \( \displaystyle (1, -1) \). Find \( \displaystyle y'' \) there.

\[ \left. 4 x^{2} + 2 y^{2} \right|_{\substack{ x=1\\ y=-1 }} = 6 \]
The point is on the curve.✓ Proved
\[ - \frac{\frac{\partial}{\partial x} \left(4 x^{2} + 2 y^{2}\right)}{\frac{\partial}{\partial y} \left(4 x^{2} + 2 y^{2}\right)} = - \frac{2 x}{y} \]
y′ = −F_x/F_y.✓ Proved
\[ \frac{d}{d x} \left(- \frac{2 x}{Y{\left(x \right)}}\right) = \frac{2 x \frac{d}{d x} Y{\left(x \right)}}{Y^{2}{\left(x \right)}} - \frac{2}{Y{\left(x \right)}} \]
Differentiate y′ again, with y a function of x.✓ Proved
Replace dy/dx in that by y′ itself, and simplify.
Reviewed
\[ \left. - \frac{4 x^{2}}{y^{3}} - \frac{2}{y} \right|_{\substack{ x=1\\ y=-1 }} = 6 \]
At (1, -1).✓ Proved

Answer \( y'' = - \frac{4 x^{2}}{y^{3}} - \frac{2}{y},\quad y''(1) = 6 \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies implicit differentiation twice and evaluates the result at the given point. The algebraic steps are verified, and the final answer is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies implicit differentiation twice and evaluates the result at the given point. The algebraic steps are verified, and the final answer is correct.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies implicit differentiation to find the second derivative and evaluates it at the given point. The intermediate steps and final result are algebraically correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-14 — 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 2026-10-14

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 2026-10-18 with SymPy 1.14.0.