Second derivatives by implicit differentiation
Problem 2.2455 · hard
The curve \( \displaystyle x^{2} + y^{2} = 5 \) passes through \( \displaystyle (-1, 2) \). Find \( \displaystyle y'' \) there.
- \[ \left. x^{2} + y^{2} \right|_{\substack{ x=-1\\ y=2 }} = 5 \]The point is on the curve.✓ Proved
- \[ - \frac{\frac{\partial}{\partial x} \left(x^{2} + y^{2}\right)}{\frac{\partial}{\partial y} \left(x^{2} + y^{2}\right)} = - \frac{x}{y} \]y′ = −F_x/F_y.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{x}{Y{\left(x \right)}}\right) = \frac{x \frac{d}{d x} Y{\left(x \right)}}{Y^{2}{\left(x \right)}} - \frac{1}{Y{\left(x \right)}} \]Differentiate y′ again, with y a function of x.✓ Proved
- Replace dy/dx in that by y′ itself, and simplify.Reviewed
- \[ \left. \frac{- x^{2} - y^{2}}{y^{3}} \right|_{\substack{ x=-1\\ y=2 }} = - \frac{5}{8} \]At (-1, 2).✓ Proved
Answer \( y'' = \frac{- x^{2} - y^{2}}{y^{3}},\quad y''(-1) = - \frac{5}{8} \)
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: passqwen3.6:27b-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 2026-10-19 — The solution correctly applies implicit differentiation twice and substitutes the point coordinates accurately.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies implicit differentiation twice and substitutes the point values accurately.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution claims y''(-1) = -5/8, but fails to substitute the y-coordinate (y=2) into the expression for y''. The correct value is -5/8 only if y=2 is used in the denominator (2^3=8), but the notation y''(-1) is ambiguous and the final step substitutes x=-1 into the formula without explicitly showing the substitution of y=2, leading to a potential confusion about whether the formula depends on y. More critically, the final answer provided in the problem statement is y''(-1) = -5/8, which is correct numerically, but the solution's step 5 notation `Subs(..., (x, y), (-1, 2))` is correct. Wait, let me re-read. The problem asks for y'' at (-1, 2). The formula is (-x^2 - y^2)/y^3. At (-1, 2), this is (-1 - 4)/8 = -5/8. The solution is actually correct. Why did I think it was wrong? Let me re-evaluate. Step 1: Point on curve. Correct. Step 2: First derivative y' = -x/y. Correct. Step 3: Differentiate y'. d/dx(-x/y) = -(y*1 - x*y')/y^2 = (-y + x(-x/y))/y^2 = (-y^2 - x^2)/y^3. The equation in step 3 is `Derivative(-x/Y(x), x) = x*Derivative(Y(x), x)/Y(x)**2 - 1/Y(x)`. Let's check this algebra. d/dx(-x/y) = - [ (1*y - x*y') / y^2 ] = -y/y^2 + x*y'/y^2 = -1/y + x*y'/y^2. This matches the RHS of step 3. Correct. Step 4: Replace y' with -x/y. So y'' = -1/y + x(-x/y)/y^2 = -1/y - x^2/y^3 = (-y^2 - x^2)/y^3. Correct. Step 5: Substitute (-1, 2). (-((-1)^2) - 2^2) / 2^3 = (-1 - 4) / 8 = -5/8. Correct. The solution is sound. The verdict should be pass.gpt-oss:20b: pass 2026-10-17
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 2026-10-19 with SymPy 1.14.0.