Home›Calculus 1›Second derivatives by implicit differentiation›Problem 2.2251

Second derivatives by implicit differentiation

Problem 2.2251 · hard

The curve \( \displaystyle x y = -4 \) passes through \( \displaystyle (-1, 4) \). Find \( \displaystyle y'' \) there.

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

Answer \( y'' = \frac{2 y}{x^{2}},\quad y''(-1) = 8 \)

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: 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 logic is sound and the steps are clear.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies implicit differentiation to find the second derivative and evaluates it at the given point. The logic is sound and the steps are clear.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-14 — Step 3 computes the derivative of -y/x as -y'/x + y/x^2, but Step 4 claims to replace dy/dx with y' to get 2y/x^2. This is algebraically incorrect; substituting y' = -y/x into -y'/x + y/x^2 yields 2y/x^2, but the text implies the derivative of -y/x is directly 2y/x^2 or that the substitution happens without the chain rule context being clear. More critically, Step 3's output is -y'/x + y/x^2. If you substitute y' = -y/x, you get -(-y/x)/x + y/x^2 = y/x^2 + y/x^2 = 2y/x^2. The sentence in Step 4 says 'Replace dy/dx ... by y' itself'. This is confusing phrasing. It should say 'Replace y' with -y/x'. As written, it suggests replacing dy/dx with the symbol y', which doesn't simplify the expression to 2y/x^2. However, the bigger issue is that Step 3 is just the quotient rule application. The transition to Step 5 skips the actual substitution of the first derivative expression. The sentence in Step 4 is misleading because it doesn't describe the necessary substitution of the expression for y' derived in Step 2. It says 'replace dy/dx ... by y' itself', which is a tautology (dy/dx is y'), not a simplification. The simplification requires replacing y' with -y/x. Therefore, the sentence is misleading/wrong about the operation performed.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The derivation of y'' is incorrect. Differentiating y' = -y/x yields y'' = -(y'x - y)/x^2. Substituting y' = -y/x gives y'' = -(-y - y)/x^2 = 2y/x^2, which matches the stated answer, but the solution text in step 3 claims the derivative of -y/x is -y'/x + y/x^2, which is correct, but step 4 says 'Replace dy/dx in that by y' itself', implying y' is replaced by -y/x. The logic is garbled and the intermediate equation in step 3 is presented as a raw derivative without showing the substitution of y' clearly. More importantly, the final result y'' = 2y/x^2 is correct, but the explanation is confusing. However, looking closer at step 3: Derivative(-Y(x)/x, x) = -Y'(x)/x + Y(x)/x^2. This is correct. Step 4 says replace dy/dx (which is Y') by y' itself? No, it should replace Y' with -y/x. The sentence is misleading/wrong because it doesn't specify what to replace it with, just 'by y' itself' which is circular or nonsensical. Actually, the biggest issue is that the solution skips the crucial substitution step in the text, making it unclear how 2y/x^2 is reached from the derivative in step 3. But wait, is the math wrong? y' = -y/x. y'' = d/dx(-y/x) = -(y'x - y)/x^2 = -((-y/x)x - y)/x^2 = -(-y - y)/x^2 = 2y/x^2. The math is correct. The sentence in step 4 is just poorly phrased. Is it an error? 'Replace dy/dx in that by y' itself' is nonsense. dy/dx IS y'. It should say 'Replace dy/dx by -y/x'. This is a misleading explanation that would confuse a student.
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.