Implicit differentiation
Problem 2.1162 · medium
The curve \( \displaystyle x^{3} + x y + y^{3} = -1 \) passes through \( \displaystyle (-1, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
- \[ -1 \]The point is on the curve.✓ Proved
- Differentiate both sides with respect to x, treating y as a function of x, then solve for dy/dx.Reviewed
- \[ \frac{d}{d x} \left(x^{3} + x Y{\left(x \right)} + Y^{3}{\left(x \right)}\right) = 3 x^{2} + \left(x + 3 Y^{2}{\left(x \right)}\right) \frac{d}{d x} Y{\left(x \right)} + Y{\left(x \right)} \]Every y term picks up a factor dy/dx.✓ Proved
- \[ \frac{- 3 x^{2} - y}{x + 3 y^{2}} \]Solve for dy/dx: minus F_x over F_y.✓ Proved
- \[ -2 \]At the point.✓ Proved
Answer \( \frac{dy}{dx} = \frac{- 3 x^{2} - y}{x + 3 y^{2}}, \quad \left.\frac{dy}{dx}\right|_{(-1,1)} = -2 \)
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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 | sympy.idiff differentiates the curve on its own and agrees |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates it at the given point. The steps are logically sound and the final answer is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates it at the given point. The steps are logically sound and the final answer is correct.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution claims the point (-1, 1) is on the curve, but substituting these values into x^3 + xy + y^3 yields -1, not -1 (wait, -1 + (-1)(1) + 1 = -1, so it is on the curve). However, the derivative calculation in line 3 is incorrect. The derivative of xy is y + x(dy/dx), but the output shows (x + 3y^2)(dy/dx) + y, which implies the derivative of x^3 was ignored or mishandled in the grouping, or rather, the term 3x^2 is present but the grouping suggests F_x = 3x^2 + y and F_y = x + 3y^2. Let's re-read line 3: '3*x**2 + (x + 3*Y(x)**2)*Derivative(Y(x), x) + Y(x)'. This expands to 3x^2 + x y' + 3y^2 y' + y. The original equation is x^3 + xy + y^3 = -1. Differentiating gives 3x^2 + (y + xy') + 3y^2 y' = 0. Rearranging: 3x^2 + y + y'(x + 3y^2) = 0. So y' = -(3x^2 + y) / (x + 3y^2). Line 4 gives (-3x^2 - y)/(x + 3y^2), which matches. The point (-1, 1) gives y' = -(3(1) + 1)/(-1 + 3) = -4/2 = -2. The math is actually correct. Why did I think it was wrong? Let me re-verify the point. (-1)^3 + (-1)(1) + 1^3 = -1 - 1 + 1 = -1. The point is on the curve. The derivative formula is correct. The evaluation is correct. The sentence in step 3 says 'Every y term picks up a factor dy/dx', which is a misleading simplification of the product rule for the xy term, but the resulting equation is correct. Is the sentence 'Every y term picks up a factor dy/dx' misleading? Yes, because the term 'y' in 'xy' does not just pick up dy/dx; the product rule applies. However, the resulting equation is correct. The prompt asks to check if sentences are misleading. Saying 'every y term picks up a factor dy/dx' is a common heuristic but technically false for mixed terms like xy where x is also a variable. But the result is right. Let's look closer. Step 1 says '-1, -1'. The problem states the point is (-1, 1). Step 1 says '-1, -1'. This is a typo in the solution text provided in the prompt? '1. [equation, unchecked] -1, -1 -- The point is on the curve.' The problem says the point is (-1, 1). The solution writes -1, -1. If the solution uses (-1, -1), let's check if (-1, -1) is on the curve: (-1)^3 + (-1)(-1) + (-1)^3 = -1 + 1 - 1 = -1. Yes, (-1, -1) is also on the curve. But the problem asks for the value at (-1, 1). If the solution calculates for (-1, -1), the answer would be different. Let's check the calculation in step 5. It says '-2'. If we use (-1, 1), we get -2. If we use (-1, -1), we get -(3(1) + (-1))/(-1 + 3(1)) = -(2)/2 = -1. The final answer is -2, which corresponds to the point (-1, 1). So the calculation in step 5 used (-1, 1). But step 1 explicitly writes '-1, -1'. This is a contradiction/typo in the setup statement. The sentence in step 1 claims the point is (-1, -1) (implied by the numbers), but the problem asks for (-1, 1). Or perhaps it's just a typo in the representation. However, the most glaring issue is usually the 'Every y term...' sentence. But the typo in the point coordinates in step 1 is an error in modeling/stating the problem data. The solution says 'The point is on the curve' next to '-1, -1'. The problem specifies '(-1, 1)'. The solution should reference (-1, 1). Using (-1, -1) in the text while calculating for (-1, 1) is confusing and technically an error in the solution's narrative. Let's look at the options. Error vs Misleading. If a student reads step 1, they see (-1, -1). They might think the problem was about (-1, -1). But the final answer -2 is for (-1, 1). This is inconsistent. I will flag this as an error.gpt-oss:20b: pass 2026-09-28
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_differentiation, checked 2026-09-28 with SymPy 1.14.0.