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The chain rule in Leibniz notation

Problem 2.2334 · easy

Let \( \displaystyle y = u^{4} \) and \( \displaystyle u = 2 x + 5 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).

\[ \frac{d}{d u} u^{4} = 4 u^{3} \]
dy/du.✓ Proved
\[ \frac{d}{d x} \left(2 x + 5\right) = 2 \]
du/dx.✓ Proved
\[ \left. 4 u^{3} \right|_{\substack{ u=2 x + 5 }} \cdot 2 = 8 \left(2 x + 5\right)^{3} \]
Multiply, and write u in terms of x.✓ Proved

Answer \( \frac{dy}{dx} = 8 \left(2 x + 5\right)^{3} \)

✓ 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: pass
qwen3.6:27b-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 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the chain rule and performs the necessary substitutions and arithmetic.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-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 2026-10-15

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.