Second derivatives by implicit differentiation
Problem 2.2253 · hard
The curve \( \displaystyle x^{3} + y^{3} = 9 \) passes through \( \displaystyle (2, 1) \). Find \( \displaystyle y'' \) there.
- \[ \left. x^{3} + y^{3} \right|_{\substack{ x=2\\ y=1 }} = 9 \]The point is on the curve.✓ Proved
- \[ - \frac{\frac{\partial}{\partial x} \left(x^{3} + y^{3}\right)}{\frac{\partial}{\partial y} \left(x^{3} + y^{3}\right)} = - \frac{x^{2}}{y^{2}} \]y′ = −F_x/F_y.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{x^{2}}{Y^{2}{\left(x \right)}}\right) = \frac{2 x^{2} \frac{d}{d x} Y{\left(x \right)}}{Y^{3}{\left(x \right)}} - \frac{2 x}{Y^{2}{\left(x \right)}} \]Differentiate y′ again, with y a function of x.✓ Proved
- Replace dy/dx in that by y′ itself, and simplify.
- \[ \left. \frac{2 x \left(- x^{3} - y^{3}\right)}{y^{5}} \right|_{\substack{ x=2\\ y=1 }} = -36 \]At (2, 1).✓ Proved
Answer \( y'' = \frac{2 x \left(- x^{3} - y^{3}\right)}{y^{5}},\quad y''(2) = -36 \)
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 derivative in line 3 is algebraically incorrect (the second term should be -2*x/y^2 * y', not -2*x/y^2). Furthermore, the final formula in line 5 is incorrect; substituting y' = -x^2/y^2 into the correct derivative yields y'' = 2x(2x^3 + y^3)/y^5, not the expression shown.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The derivative in line 3 is algebraically incorrect (the second term should be -2*x/y^2 * y', not -2*x/y^2). Furthermore, the final formula in line 5 is incorrect; substituting y' = -x^2/y^2 into the correct derivative yields y'' = 2x(2x^3 + y^3)/y^5, not the expression shown.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-14 — The differentiation in step 3 is incorrect; the derivative of -x^2 y^{-2} is -2x y^{-2} + 2x^2 y^{-3} y', but the solution writes 2x^2 y' y^{-3} - 2x y^{-2}, missing a factor of x in the second term (it should be -2x/y^2, not -2x/y^2? Wait, d/dx(-x^2 y^{-2}) = -2x y^{-2} - x^2(-2)y^{-3}y' = -2x/y^2 + 2x^2 y'/y^3. The solution has -2x/y^2. Let's re-read carefully. Step 3 says: 2*x**2*y'/y**3 - 2*x/y**2. This matches -2x/y^2 + 2x^2 y'/y^3. So the differentiation is actually correct. Let's check the simplification. y' = -x^2/y^2. Substitute into 2x^2(-x^2/y^2)/y^3 - 2x/y^2 = -2x^4/y^5 - 2x/y^2 = (-2x^4 - 2xy^3)/y^5. Since x^3+y^3=9, y^3 = 9-x^3. So -2x^4 - 2x(9-x^3) = -2x^4 - 18x + 2x^4 = -18x. So y'' = -18x/y^5. The stated answer is 2x(-x^3-y^3)/y^5 = -2x(9)/y^5 = -18x/y^5. The formula in step 5 is 2*x*(-x**3 - y**3)/y**5. This equals -18x/y^5. The value at (2,1) is -18(2)/1 = -36. The math is correct. Why did I think it was wrong? Let's re-read step 3. Derivative(-x**2/Y(x)**2, x) = 2*x**2*Derivative(Y(x), x)/Y(x)**3 - 2*x/Y(x)**2. Product rule on -x^2 * y^{-2}. d/dx(-x^2) = -2x. d/dx(y^{-2}) = -2y^{-3}y'. So (-2x)(y^{-2}) + (-x^2)(-2y^{-3}y') = -2x/y^2 + 2x^2 y'/y^3. The solution has 2x^2 y'/y^3 - 2x/y^2. This is correct. The simplification in step 4/5 leads to the correct result. The verdict should be pass.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: fail (error) 2026-10-14 — The differentiation in line 3 is incorrect; the derivative of -x^2/y^2 with respect to x is -2x/y^2 + 2x^2 y'/y^3, but the solution writes 2x^2 y'/y^3 - 2x/y^2 (missing a factor of x in the second term's numerator relative to the first, or rather, the term -2x/y^2 is correct for d/dx(-x^2/y^2) only if y is constant, but the chain rule application is messy and the resulting simplified expression in line 5 is algebraically inconsistent with the correct second derivative formula for this curve. Specifically, y'' = -2x/y^2 + 2x^2(-x^2/y^2)/y^3 = -2x/y^2 - 2x^4/y^5. The solution's numerator 2x(-x^3-y^3) = -2x^4 - 2xy^3, which divided by y^5 gives -2x^4/y^5 - 2x/y^2. This matches. However, line 3's expression `2*x**2*Derivative(Y(x), x)/Y(x)**3 - 2*x/Y(x)**2` is actually correct for the derivative of `-x^2/y^2`? Let's check: d/dx(-x^2 y^-2) = -2x y^-2 + (-x^2)(-2)y^-3 y' = -2x/y^2 + 2x^2 y'/y^3. Line 3 says `2*x**2*y'/y**3 - 2*x/y**2`. This is correct. Wait, let's re-read line 3 carefully. `Derivative(-x**2/Y(x)**2, x) = 2*x**2*Derivative(Y(x), x)/Y(x)**3 - 2*x/Y(x)**2`. This is correct. Then line 4 says replace y'. Line 5 substitutes y' = -x^2/y^2 into the expression. 2x^2(-x^2/y^2)/y^3 - 2x/y^2 = -2x^4/y^5 - 2x/y^2 = (-2x^4 - 2xy^3)/y^5 = 2x(-x^3-y^3)/y^5. Since x^3+y^3=9, this is 2x(-9)/y^5 = -18x/y^5. At (2,1), -18(2)/1 = -36. The math is actually correct. Why did I think it was wrong? Let me re-evaluate. Re-evaluation: 1. Point check: 8+1=9. Correct. 2. y' = -x^2/y^2. Correct. 3. y'' = d/dx(-x^2 y^-2) = -2x y^-2 + (-x^2)(-2)y^-3 y' = -2x/y^2 + 2x^2 y'/y^3. Line 3 writes `2*x**2*y'/y**3 - 2*x/y**2`. This is correct. 4. Substitute y'. 5. Result: 2x^2(-x^2/y^2)/y^3 - 2x/y^2 = -2x^4/y^5 - 2x/y^2 = (-2x^4 - 2xy^3)/y^5 = 2x(-x^3-y^3)/y^5. Correct. 6. Value: 2(2)(-9)/1 = -36. Correct. The solution is actually correct. I will mark it as pass.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.