The chain rule in Leibniz notation
Problem 2.2223 · easy
Let \( \displaystyle y = \sin{\left(u \right)} \) and \( \displaystyle u = 2 x + 2 \). 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
- \[ \frac{d}{d x} \left(2 x + 2\right) = 2 \]du/dx.✓ Proved
- \[ \left. \cos{\left(u \right)} \right|_{\substack{ u=2 x + 2 }} \cdot 2 = 2 \cos{\left(2 x + 2 \right)} \]Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = 2 \cos{\left(2 x + 2 \right)} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. 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, computes the partial derivatives, and substitutes back to express the result in terms of x.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the chain rule, computes the partial derivatives, and substitutes back to express the result in terms of x.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the chain rule, computes the partial derivatives, and substitutes the inner function back into the outer derivative.gpt-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.