The chain rule in Leibniz notation
Problem 2.2306 · easy
Let \( \displaystyle y = \sin{\left(u \right)} \) and \( \displaystyle u = 4 - x^{2} \). 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
- \[ \frac{d}{d x} \left(4 - x^{2}\right) = - 2 x \]du/dx.✓ Proved
- \[ - 2 x \left. \cos{\left(u \right)} \right|_{\substack{ u=4 - x^{2} }} = - 2 x \cos{\left(x^{2} - 4 \right)} \]Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = - 2 x \cos{\left(x^{2} - 4 \right)} \)
✓ 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 and performs the substitution. The final expression is mathematically equivalent to the stated answer since cosine is an even function.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the chain rule and performs the substitution. The final expression is mathematically equivalent to the stated answer since cosine is an even function.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the chain rule and substitutes the inner function. Although the argument of the cosine is written as x^2 - 4 instead of 4 - x^2, this is mathematically correct because cosine is an even function.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the chain rule and performs the substitution. Although the argument of the cosine is written as x^2 - 4 instead of 4 - x^2, this is mathematically valid because cosine is an even function.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.