Implicit differentiation

Reviewers

• gpt-oss:20b: pass
• qwen3.6:27b-mlx: fail (error) — The solution incorrectly identifies the derivative of the constant right-hand side as non-zero in the intermediate step (Line 3), effectively treating the RHS as if it were part of the LHS or failing to set the total derivative to zero. Specifically, Line 3 shows the derivative of the LHS equal to an expression that includes terms from the LHS derivative but implies the RHS derivative was ignored or mishandled in the setup description, although the final algebraic result in Line 4 is correct. However, the sentence in Line 2 is vague, and Line 3's equation is structurally confusing: it equates the derivative of the LHS to a mix of terms that looks like the numerator of the final answer plus extra terms, rather than clearly showing d/dx(LHS) = d/dx(RHS) = 0. More critically, Line 3 claims the derivative of the LHS equals `(x*cos(x*y) + 3)*dy/dx + y*cos(x*y) - 3`. Let's check: d/dx(-3x + 3y + sin(xy)) = -3 + 3y' + cos(xy)(y + xy'). This equals y'(3 + x cos(xy)) + (y cos(xy) - 3). The RHS of the original equation is constant, so its derivative is 0. Line 3 sets the LHS derivative equal to `...`. It does not explicitly state it equals 0. But the bigger issue is that Line 3 is marked 'proved' by a CAS, so the algebra is likely correct, but the *sentence* in Line 2 says 'Differentiate both sides... then solve'. Line 3 shows the differentiation of the LHS. It does not show the RHS differentiation (which is 0). It just shows an identity for the LHS derivative. Then Line 4 jumps to the solved form. The transition from 3 to 4 skips the step of setting the derivative to 0. While not a 'false' statement per se, it's a non-sequitur in the logical flow if Line 3 is just an expansion. However, looking closely at Line 3: `Derivative(...) = ...`. The RHS of Line 3 is `y'(x cos + 3) + y cos - 3`. This is indeed the expanded derivative. The next step should be `... = 0`. Line 4 is the solution for y'. The missing step is setting it to 0. Is this an error? It's a gap. But wait, Line 1 checks the point. Line 3 expands the derivative. Line 4 solves for dy/dx. The implicit assumption is that the derivative of the constant is 0. This is standard. Let's look closer at Line 3's RHS: `(x*cos(x*Y(x)) + 3)*Derivative(Y(x), x) + Y(x)*cos(x*Y(x)) - 3`. Actual derivative: `-3 + 3y' + cos(xy)(y + xy') = -3 + 3y' + y cos(xy) + x y' cos(xy) = y'(3 + x cos(xy)) + (y cos(xy) - 3)`. This matches Line 3's RHS. So Line 3 is `d(LHS)/dx = ExpandedForm`. Line 4 is `dy/dx = SolvedForm`. The logical link is `ExpandedForm = 0`. This is missing. However, usually 'error' is reserved for incorrect math. Is skipping ' = 0' an error? It's a missing step. Let's re-read the prompt constraints. 'Check that... each conclusion follows from the lines before it'. Does Line 4 follow from Line 3? No. Line 3 is an identity `A = B`. Line 4 is `y' = C`. You cannot get `y' = C` from `A = B` without knowing `A = 0` (or `B = 0`). The fact that the RHS of the original equation is constant is not explicitly used in the transition from 3 to 4. Therefore, the conclusion does not follow from the previous lines alone. It requires the external knowledge that the derivative of the constant RHS is 0. While this is trivial, strictly speaking, the logical chain is broken. Also, Line 1 is `6 - sin(1) = 6 - sin(1)`. This is a tautology, not a verification that the point is on the curve. To verify the point is on the curve, one should substitute x=-1, y=1 into the LHS and show it equals the RHS. Line 1 just states the RHS equals itself. This is a 'misleading' or 'style' issue, or potentially an 'error' in the verification step if it's meant to be a check. The problem states the curve passes through the point, so verification might not be strictly necessary, but Line 1 claims to do it and fails to actually substitute the values. Let's look at Line 1 again: `6 - sin(1) = 6 - sin(1)`. This is not checking if (-1,1) is on the curve. It's just writing the constant. This is a failed verification step. Verdict: fail. Severity: error (because the verification step is logically empty/wrong, and the differentiation step skips the crucial '=0' link). Actually, the most glaring issue is Line 1. It claims to verify the point is on the curve but performs no substitution. It just writes the constant term. This is incorrect procedure. Also, Line 3 to 4 gap. I will cite the verification failure as the primary error.

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 2026-10-03 with SymPy 1.14.0.