Home›Calculus 1›The chain rule in Leibniz notation›Problem 2.2437

The chain rule in Leibniz notation

Problem 2.2437 · easy

Let \( \displaystyle y = u^{2} - u \) and \( \displaystyle u = \cos{\left(x \right)} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).

\[ \frac{d}{d u} \left(u^{2} - u\right) = 2 u - 1 \]
dy/du.✓ Proved
\[ \frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)} \]
du/dx.✓ Proved
\[ - \sin{\left(x \right)} \left. 2 u - 1 \right|_{\substack{ u=\cos{\left(x \right)} }} = \sin{\left(x \right)} - \sin{\left(2 x \right)} \]
Multiply, and write u in terms of x.✓ Proved

Answer \( \frac{dy}{dx} = \sin{\left(x \right)} - \sin{\left(2 x \right)} \)

Every line of this solution was proved by the computer algebra system SymPy. 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
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	a central difference of y(u(x)) at three points

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the chain rule, computes the derivatives, and substitutes back to express the result in terms of x. The final simplification is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies the chain rule, computes the derivatives, and substitutes back to express the result in terms of x. The final simplification is correct.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The algebraic simplification in step 3 is incorrect. The product (-sin(x))(2cos(x) - 1) simplifies to sin(x) - 2sin(x)cos(x), which equals sin(x) - sin(2x). However, the solution claims the result is sin(x) - sin(2x) directly from the multiplication without showing the factor of 2, or implies (-sin(x))*(2cos(x)-1) = sin(x) - sin(2x) is a direct identity which is false (it misses the factor of 2 on the second term if interpreted as -2sin(x)cos(x) = -sin(2x), wait: -2sin(x)cos(x) IS -sin(2x). Let me re-evaluate. Step 3 says: (-sin(x)) * (2cos(x) - 1) = sin(x) - sin(2x). LHS: -2sin(x)cos(x) + sin(x). RHS: sin(x) - sin(2x) = sin(x) - 2sin(x)cos(x). They are equal. Wait, let me look closer at the provided solution text. Line 3: `(-sin(x))*Subs(2*u - 1, u, cos(x)) = sin(x) - sin(2*x)` LHS: `-sin(x) * (2cos(x) - 1) = -2sin(x)cos(x) + sin(x)`. RHS: `sin(x) - sin(2x) = sin(x) - 2sin(x)cos(x)`. These are identical. The algebra is correct. Why did I think it was wrong? I misread the sign or the factor. Let's check the stated answer: `sin(x) - sin(2x)`. Let's check the derivative manually. y = cos^2(x) - cos(x). dy/dx = 2cos(x)(-sin(x)) - (-sin(x)) = -2sin(x)cos(x) + sin(x) = sin(x) - sin(2x). The solution is correct. So why would I fail it? Is there a subtle error? "Multiply, and write u in terms of x." The equation is marked [equation, proved]. The prompt says "do not re-check the algebra of a line marked proved". So if the algebra is correct, and the setup is correct, it should pass. Let's double check the prompt constraints. "Each line ... is either an EQUATION ... or a SENTENCE ... The sentences are what you are here for." Line 3 is an equation. It is marked proved. Are there any sentences? No explicit sentences like "By the chain rule...". The comments `-- dy/du` etc are part of the equation line metadata or comments. Is the setup correct? Problem: Find dy/dx using dy/dx = dy/du * du/dx. Step 1: dy/du. Correct. Step 2: du/dx. Correct. Step 3: Multiply and substitute. Correct. It seems sound. Wait, look at the Stated Answer vs Solution Step 3. Stated answer: `\frac{dy}{dx} = \sin{(x)} - \sin{(2 x)}` Solution Step 3 RHS: `sin(x) - sin(2*x)` They match. Is there any ambiguity? Maybe the order of terms? No, addition is commutative. I will mark it as pass.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The final line claims that (-sin(x))(2cos(x) - 1) equals sin(x) - sin(2x). This is algebraically incorrect; the correct expansion is sin(x) - 2sin(x
gpt-oss:20b: pass 2026-10-16

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/chain_rule_leibniz, checked 2026-10-19 with SymPy 1.14.0.