Home›Calculus 1›The chain rule in Leibniz notation›Problem 2.2303

The chain rule in Leibniz notation

Problem 2.2303 · easy

Let \( \displaystyle y = - 2 u - 4 \) and \( \displaystyle u = - 3 x^{2} - 1 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).

\[ \frac{d}{d u} \left(- 2 u - 4\right) = -2 \]
dy/du.✓ Proved
\[ \frac{d}{d x} \left(- 3 x^{2} - 1\right) = - 6 x \]
du/dx.✓ Proved
\[ - 6 x \left. -2 \right|_{\substack{ u=- 3 x^{2} - 1 }} = 12 x \]
Multiply, and write u in terms of x.✓ Proved

Answer \( \frac{dy}{dx} = 12 x \)

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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the chain rule and computes the derivatives. The final expression is in terms of x as required.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the chain rule and computes the derivatives. The final expression is in terms of x as required.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the chain rule and computes the derivatives. Although the notation in step 3 is slightly unconventional (suggesting substitution into a constant), the arithmetic result is correct and the logic holds.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution incorrectly substitutes the constant derivative dy/du into the expression for u, implying a dependency that does not exist. The chain rule requires multiplying dy/du by du/dx directly, not substituting dy/du into u.
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 2026-10-18 with SymPy 1.14.0.