The chain rule in Leibniz notation
Problem 2.2304 · easy
Let \( \displaystyle y = - 4 u - 4 \) and \( \displaystyle u = x^{2} + 1 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
- \[ \frac{d}{d u} \left(- 4 u - 4\right) = -4 \]dy/du.✓ Proved
- \[ \frac{d}{d x} \left(x^{2} + 1\right) = 2 x \]du/dx.✓ Proved
- \[ 2 x \left. -4 \right|_{\substack{ u=x^{2} + 1 }} = - 8 x \]Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = - 8 x \)
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, computes the derivatives, and substitutes back to express the result in terms of x.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the chain rule, computes the derivatives, and substitutes back to express the result in terms of x.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the chain rule, computes the individual derivatives, and substitutes back to express the result in terms of x.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the chain rule, computes the derivatives accurately, and substitutes back to express the result in terms of x.gpt-oss:20b: fail (misleading) 2026-10-14 — Step 3 incorrectly uses a substitution of -4 for u; the correct multiplication is (2x)*(-4) = -8x. The substitution shown is unnecessary and could confuse a student.
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.