Home›Calculus 1›Second derivatives by implicit differentiation›Problem 2.2327

Second derivatives by implicit differentiation

Problem 2.2327 · hard

The curve \( \displaystyle 2 x^{2} + 4 y^{2} = 18 \) passes through \( \displaystyle (-1, 2) \). Find \( \displaystyle y'' \) there.

\[ \left. 2 x^{2} + 4 y^{2} \right|_{\substack{ x=-1\\ y=2 }} = 18 \]
The point is on the curve.✓ Proved
\[ - \frac{\frac{\partial}{\partial x} \left(2 x^{2} + 4 y^{2}\right)}{\frac{\partial}{\partial y} \left(2 x^{2} + 4 y^{2}\right)} = - \frac{x}{2 y} \]
y′ = −F_x/F_y.✓ Proved
\[ \frac{d}{d x} \left(- \frac{x}{2 Y{\left(x \right)}}\right) = \frac{x \frac{d}{d x} Y{\left(x \right)}}{2 Y^{2}{\left(x \right)}} - \frac{1}{2 Y{\left(x \right)}} \]
Differentiate y′ again, with y a function of x.✓ Proved
Replace dy/dx in that by y′ itself, and simplify.
\[ \left. \frac{- x^{2} - 2 y^{2}}{4 y^{3}} \right|_{\substack{ x=-1\\ y=2 }} = - \frac{9}{32} \]
At (-1, 2).✓ Proved

Answer \( y'' = \frac{- x^{2} - 2 y^{2}}{4 y^{3}},\quad y''(-1) = - \frac{9}{32} \)

Lines: 4 proved, 1 not checked. 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	Not checked	—	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: fail (error) — The derivation of y'' in step 3 is algebraically incorrect; the quotient rule was applied incorrectly, missing the chain rule factor for the denominator's derivative and resulting in the wrong sign for the second term. The correct derivative of -x/(2y) is (1 - x y')/(2y^2), not the expression shown.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The derivation of y'' in step 3 is algebraically incorrect; the quotient rule was applied incorrectly, missing the chain rule factor for the denominator's derivative and resulting in the wrong sign for the second term. The correct derivative of -x/(2y) is (1 - x y')/(2y^2), not the expression shown.
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 algebraic steps are verified, and the logic is sound.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to substitute the known relationship 2x^2 + 4y^2 = 18 into the numerator of the second derivative expression. While the algebraic form y'' = (-x^2 - 2y^2)/(4y^3) is correct, standard practice requires simplifying the numerator to -9/2 using the original equation before evaluating, or at least acknowledging that the numerator is constant on the curve. More critically, the prompt asks for y'' *there*, and while the final numerical answer is correct, the intermediate symbolic expression provided in the 'Stated answer' part of the prompt is not fully simplified, but the solution steps don't explicitly show the substitution of the curve equation to simplify the numerator, which is a key part of solving such problems efficiently and correctly. However, looking closely, the solution just computes the value. Is it wrong? No. Is it misleading? The stated answer includes the unsimplified formula. The solution steps are technically correct but miss the simplification step that makes the evaluation robust. Wait, let's look at step 4. 'Replace dy/dx... and simplify'. Step 5 substitutes values. The result is correct. Is there an error? Let's re-read carefully. Step 2: y' = -x/(2y). Correct. Step 3: Differentiates -x/(2y). d/dx(-x/(2y)) = - [ (1)(2y) - x(2y') ] / (2y)^2 = - [ 2y - 2xy' ] / 4y^2 = (2xy' - 2y) / 4y^2 = (xy' - y) / 2y^2. Let's check the equation in step 3: `x*Derivative(Y(x), x)/(2*Y(x)**2) - 1/(2*Y(x))`. This equals `xy'/(2y^2) - 1/(2y) = (xy' - y)/2y^2`. This matches. Step 4: Replace y'. y'' = (x(-x/2y) - y) / 2y^2 = (-x^2/2y - y) / 2y^2 = (-x^2 - 2y^2) / 4y^3. Correct. Step 5: Substitute x=-1, y=2. Numerator: -(-1)^2 - 2(2)^2 = -1 - 8 = -9. Denominator: 4(2)^3 = 32. Result -9/32. Correct. The solution is mathematically sound. The 'Stated answer' in the problem description includes the formula, which is fine. The solution derives it and evaluates it. There is no error. Wait, did I miss something? Problem: Find y'' there. Solution finds it. Is there a subtle issue? Maybe the notation `Y(x)` in step 3 is slightly inconsistent with `y` in step 2, but it's clear. Maybe the fact that it doesn't use 2x^2+4y^2=18 to simplify the numerator to -9/2 is considered 'style' or 'misleading' if the student thinks the numerator varies? But the final answer is a number. Let's look at the 'Stated answer' again: `y'' = \frac{- x^{2} - 2 y^{2}}{4 y^{3}},\quad y''(-1) = - \frac{9}{32}`. The solution produces this. Is there any reason to fail? If I am adversarial: The solution does not explicitly state that 2x^2 + 4y^2 = 18 implies -x^2 - 2y^2 = -9/2. It just plugs in numbers. This is valid. However, often in these problems, the 'trap' is forgetting that y is a function of x. Step 3 handles this. Let's check the sign in Step 3 again. `Derivative(-x/(2*Y(x)), x)` Quotient rule: `(u'v - uv')/v^2`. u = -x, u' = -1. v = 2Y, v' = 2Y'. `(-1 * 2Y - (-x) * 2Y') / (2Y)^2` `= (-2Y + 2xY') / 4Y^2` `= (xY' - Y) / 2Y^2` `= xY'/(2Y^2) - Y/(2Y^2)` `= xY'/(2Y^2) - 1/(2Y)`. Step 3 says: `x*Derivative(Y(x), x)/(2*Y(x)**2) - 1/(2*Y(x))`. This is correct. I see no error. I will pass it.
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.