Home›Calculus 1›Second derivatives by implicit differentiation›Problem 2.2373

Second derivatives by implicit differentiation

Problem 2.2373 · hard

The curve \( \displaystyle 4 x^{2} + y^{2} = 8 \) passes through \( \displaystyle (-1, 2) \). Find \( \displaystyle y'' \) there.

\[ \left. 4 x^{2} + y^{2} \right|_{\substack{ x=-1\\ y=2 }} = 8 \]
The point is on the curve.✓ Proved
\[ - \frac{\frac{\partial}{\partial x} \left(4 x^{2} + y^{2}\right)}{\frac{\partial}{\partial y} \left(4 x^{2} + y^{2}\right)} = - \frac{4 x}{y} \]
y′ = −F_x/F_y.✓ Proved
\[ \frac{d}{d x} \left(- \frac{4 x}{Y{\left(x \right)}}\right) = \frac{4 x \frac{d}{d x} Y{\left(x \right)}}{Y^{2}{\left(x \right)}} - \frac{4}{Y{\left(x \right)}} \]
Differentiate y′ again, with y a function of x.✓ Proved
Replace dy/dx in that by y′ itself, and simplify.
Reviewed
\[ \left. - \frac{16 x^{2}}{y^{3}} - \frac{4}{y} \right|_{\substack{ x=-1\\ y=2 }} = -4 \]
At (-1, 2).✓ Proved

Answer \( y'' = - \frac{16 x^{2}}{y^{3}} - \frac{4}{y},\quad y''(-1) = -4 \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies implicit differentiation twice and substitutes the point coordinates accurately.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies implicit differentiation twice and substitutes the point coordinates accurately.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies implicit differentiation twice and substitutes the point coordinates accurately.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — 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 logic is sound.
gpt-oss:20b: pass 2026-10-15

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.