Home›Calculus 1›The chain rule in Leibniz notation›Problem 2.2345

The chain rule in Leibniz notation

Problem 2.2345 · easy

Let \( \displaystyle y = \sin{\left(u \right)} \) and \( \displaystyle u = - 2 x^{2} - 4 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).

\[ \frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)} \]
dy/du.✓ Proved
\[ \frac{d}{d x} \left(- 2 x^{2} - 4\right) = - 4 x \]
du/dx.✓ Proved
\[ - 4 x \left. \cos{\left(u \right)} \right|_{\substack{ u=- 2 x^{2} - 4 }} = - 4 x \cos{\left(2 x^{2} + 4 \right)} \]
Multiply, and write u in terms of x.✓ Proved

Answer \( \frac{dy}{dx} = - 4 x \cos{\left(2 x^{2} + 4 \right)} \)

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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The final answer is incorrect. Since cos(u) is an even function, cos(-2x^2 - 4) equals cos(2x^2 + 4), but the derivative calculation yields -4x * co

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The final answer is incorrect. Since cos(u) is an even function, cos(-2x^2 - 4) equals cos(2x^2 + 4), but the derivative calculation yields -4x * co
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The final answer is incorrect. Since cos(u) is an even function, cos(-2x^2 - 4) equals cos(2x^2 + 4), but the solution drops the negative sign insid
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The final equation in step 3 is algebraically incorrect. Substituting u = -2x^2 - 4 into cos(u) yields cos(-2x^2 - 4), which simplifies to cos(2x^2
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/chain_rule_leibniz, checked 2026-10-18 with SymPy 1.14.0.