Second derivatives by implicit differentiation
Problem 2.2365 · hard
The curve \( \displaystyle 4 x^{2} + 3 y^{2} = 7 \) passes through \( \displaystyle (1, -1) \). Find \( \displaystyle y'' \) there.
- \[ \left. 4 x^{2} + 3 y^{2} \right|_{\substack{ x=1\\ y=-1 }} = 7 \]The point is on the curve.✓ Proved
- \[ - \frac{\frac{\partial}{\partial x} \left(4 x^{2} + 3 y^{2}\right)}{\frac{\partial}{\partial y} \left(4 x^{2} + 3 y^{2}\right)} = - \frac{4 x}{3 y} \]y′ = −F_x/F_y.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{4 x}{3 Y{\left(x \right)}}\right) = \frac{4 x \frac{d}{d x} Y{\left(x \right)}}{3 Y^{2}{\left(x \right)}} - \frac{4}{3 Y{\left(x \right)}} \]Differentiate y′ again, with y a function of x.✓ Proved
- Replace dy/dx in that by y′ itself, and simplify.
- \[ \left. \frac{4 \left(- 4 x^{2} - 3 y^{2}\right)}{9 y^{3}} \right|_{\substack{ x=1\\ y=-1 }} = \frac{28}{9} \]At (1, -1).✓ Proved
Answer \( y'' = \frac{4 \left(- 4 x^{2} - 3 y^{2}\right)}{9 y^{3}},\quad y''(1) = \frac{28}{9} \)
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: passqwen3.6:27b-mlx: fail (error) — The solution fails to substitute the original equation 4x^2 + 3y^2 = 7 into the expression for y'' before evaluating at the point. The formula derived in step 5 contains (-4x^2 - 3y^2), which equals -7 on the curve, but the evaluation in step 5 substitutes x=1, y=-1 directly into the unsimplified expression, yielding 4*(-4-3)/(9*-1) = 28/9. While the numerical answer happens to be correct because the point satisfies the curve equation, the stated general formula for y'' is incorrect (it should be 28/(9y^3) or similar, not containing the variable terms that sum to a constant). More critically, step 3's differentiation is messy and step 4 is vague, but the main error is presenting a non-simplified, variable-dependent formula as the final answer for y'' when it can and should be simplified using the constraint. Actually, looking closer: y'' = (4x y' - 4y)/ (3y^2) ... wait. Let's re-derive. 8x + 6yy' = 0 => y' = -4x/3y. y'' = - (3y(4) - 4x(3y')) / (9y^2) = - (12y - 12xy') / (9y^2) = -4(3y - 3xy') / (9y^2) = -4/3y^2 * (y - xy'). Substitute y': y - x(-4x/3y) = y + 4x^2/3y = (3y^2 + 4x^2)/3y = 7/3y. So y'' = -4/(3y^2) * (7/3y) = -28/(9y^3). At (1, -1), y'' = -28/(9(-1)) = 28/9. The solution's formula is 4(-4x^2 - 3y^2)/(9y^3). Since 4x^2+3y^2=7, -4x^2-3y^2 = -7. So the formula gives 4(-7)/(9y^3) = -28/(9y^3). This matches my derivation. So the formula is technically correct if one assumes the point is on the curve, but it's bad practice to leave the constraint in the numerator. However, the verdict is 'fail' because the sentence in step 4 says 'simplify' but the result in step 5 is not fully simplified with respect to the constraint, and more importantly, the prompt asks to review the *sentences*. Step 4 is vague. But is it an error? The formula in step 5 is algebraically equivalent to the correct y'' on the curve. The main issue is that the solution presents a formula for y'' that depends on x and y, which is fine, but it doesn't explicitly state that 4x^2+3y^2 is replaced by 7. The evaluation in step 5 works numerically. Is there a logical gap? Step 3 differentiates -4x/(3Y(x)). The result is 4x Y' / (3Y^2) - 4/(3Y). This is correct. Step 4 says replace dy/dx by y'. y' is -4x/3y. So y'' = 4x(-4x/3y)/(3y^2) - 4/(3y) = -16x^2/(9y^3) - 12y^2/(9y^3) = (-16x^2 - 12y^2)/(9y^3) = -4(4x^2+3y^2)/(9y^3). The solution has 4(-4x^2-3y^2)/(9y^3) which is the same. So the math is correct. Why would it fail? Perhaps the 'sentence' in step 4 is too brief? Or is there a sign error? My derivation: y'' = -28/(9y^3). Solution: 4(-7)/(9y^3) = -28/(9y^3). They match. The evaluation 28/9 is correct. The solution is sound. Wait, look at step 3 again. Derivative of -4x/(3y). Quotient rule: ( (-4)(3y) - (-4x)(3y') ) / (3y)^2 = (-12y + 12xy') / 9y^2. Step 3 says: 4x y' / (3y^2) - 4/(3y). Let's check: 4x y' / (3y^2) = 12x y' / 9y^2. -4/(3y) = -12y / 9y^2. Sum: (12xy' - 12y)/9y^2. This matches. So the algebra is correct. The sentences are minimal but not wrong. Verdict: pass.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to substitute the original equation 4x^2 + 3y^2 = 7 into the expression for y'' before evaluating at the point. The formula derived in step 5 contains (-4x^2 - 3y^2), which equals -7 on the curve, but the evaluation in step 5 substitutes x=1, y=-1 directly into the unsimplified expression, yielding 4*(-4-3)/(9*-1) = 28/9. While the numerical answer happens to be correct because the point satisfies the curve equation, the stated general formula for y'' is incorrect (it should be 28/(9y^3) or similar, not containing the variable terms that sum to a constant). More critically, step 3's differentiation is messy and step 4 is vague, but the main error is presenting a non-simplified, variable-dependent formula as the final answer for y'' when it can and should be simplified using the constraint. Actually, looking closer: y'' = (4x y' - 4y)/ (3y^2) ... wait. Let's re-derive. 8x + 6yy' = 0 => y' = -4x/3y. y'' = - (3y(4) - 4x(3y')) / (9y^2) = - (12y - 12xy') / (9y^2) = -4(3y - 3xy') / (9y^2) = -4/3y^2 * (y - xy'). Substitute y': y - x(-4x/3y) = y + 4x^2/3y = (3y^2 + 4x^2)/3y = 7/3y. So y'' = -4/(3y^2) * (7/3y) = -28/(9y^3). At (1, -1), y'' = -28/(9(-1)) = 28/9. The solution's formula is 4(-4x^2 - 3y^2)/(9y^3). Since 4x^2+3y^2=7, -4x^2-3y^2 = -7. So the formula gives 4(-7)/(9y^3) = -28/(9y^3). This matches my derivation. So the formula is technically correct if one assumes the point is on the curve, but it's bad practice to leave the constraint in the numerator. However, the verdict is 'fail' because the sentence in step 4 says 'simplify' but the result in step 5 is not fully simplified with respect to the constraint, and more importantly, the prompt asks to review the *sentences*. Step 4 is vague. But is it an error? The formula in step 5 is algebraically equivalent to the correct y'' on the curve. The main issue is that the solution presents a formula for y'' that depends on x and y, which is fine, but it doesn't explicitly state that 4x^2+3y^2 is replaced by 7. The evaluation in step 5 works numerically. Is there a logical gap? Step 3 differentiates -4x/(3Y(x)). The result is 4x Y' / (3Y^2) - 4/(3Y). This is correct. Step 4 says replace dy/dx by y'. y' is -4x/3y. So y'' = 4x(-4x/3y)/(3y^2) - 4/(3y) = -16x^2/(9y^3) - 12y^2/(9y^3) = (-16x^2 - 12y^2)/(9y^3) = -4(4x^2+3y^2)/(9y^3). The solution has 4(-4x^2-3y^2)/(9y^3) which is the same. So the math is correct. Why would it fail? Perhaps the 'sentence' in step 4 is too brief? Or is there a sign error? My derivation: y'' = -28/(9y^3). Solution: 4(-7)/(9y^3) = -28/(9y^3). They match. The evaluation 28/9 is correct. The solution is sound. Wait, look at step 3 again. Derivative of -4x/(3y). Quotient rule: ( (-4)(3y) - (-4x)(3y') ) / (3y)^2 = (-12y + 12xy') / 9y^2. Step 3 says: 4x y' / (3y^2) - 4/(3y). Let's check: 4x y' / (3y^2) = 12x y' / 9y^2. -4/(3y) = -12y / 9y^2. Sum: (12xy' - 12y)/9y^2. This matches. So the algebra is correct. The sentences are minimal but not wrong. Verdict: pass.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-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 final result is correct.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-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 final result is correct.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.