The chain rule in Leibniz notation
Problem 2.2218 · easy
Let \( \displaystyle y = \ln{\left(u \right)} \) and \( \displaystyle u = \cos{\left(x \right)} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
- \[ \frac{d}{d u} \ln{\left(u \right)} = \frac{1}{u} \]dy/du.✓ Proved
- \[ \frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)} \]du/dx.✓ Proved
- \[ - \sin{\left(x \right)} \left. 1 \frac{1}{u} \right|_{\substack{ u=\cos{\left(x \right)} }} = - \tan{\left(x \right)} \]Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = - \tan{\left(x \right)} \)
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: passqwen3.6:27b-mlx: pass — The solution correctly applies the chain rule and simplifies the resulting expression. The steps are logically sound and algebraically correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the chain rule and simplifies the resulting expression. The steps are logically sound and algebraically correct.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the chain rule and simplifies the expression. The steps are logically sound and the final result is correct.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The final step contains an algebraic error: -sin(x)/cos(x) is equal to -tan(x), but the solution implies a multiplication or substitution that resulgpt-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/chain_rule_leibniz, checked 2026-10-17 with SymPy 1.14.0.