Derivative of \( \displaystyle - \cos{\left(x - 1 \right)} \)
Problem 2.106 · medium
Differentiate \( \displaystyle f(x) = - \cos{\left(x - 1 \right)} \).
- \[ \frac{d}{d x} \left(- \cos{\left(x - 1 \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} \cos{\left(x - 1 \right)} \]constantPull the constant factor -1 out of the derivative.✓ Proved
- \[ = \sin{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right) \]trig algebraApply the derivative rule for cosine. Simplify the signs.✓ Proved
- \[ = \sin{\left(x - 1 \right)} \]derivative algebraDifferentiate the inner function x - 1. Simplify the final expression.✓ Proved
Answer \( \sin{\left(x - 1 \right)} \)
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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: fail (style) — Step 1 uses an unlabelled rule; it should be identified as a derivative step. All other steps are correctly labeled and apply only one rule each.deepseek-r1:70b: fail (style) — Step 1 is missing a rule label, which is required by the contract.qwen3.6:27b-mlx: pass
Every verdict on record (15)
qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (style) 2026-09-20 — Step 1 uses an unlabelled rule; it should be identified as a derivative step. All other steps are correctly labeled and apply only one rule each.qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (error) 2026-09-20 — Step 3 applies two rules at once: it uses the derivative of cosine (trig) and the chain rule for the inner function x‑1. It should be split into separate steps, each labeled with a single rule from the allowed vocabulary.qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: fail (style) 2026-09-19 — Step 1 is missing a rule label, which is required by the contract.gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: fail (style) 2026-09-19 — The first step lacks a rule label, which is required by the contract.gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18deepseek-r1:70b: fail (error) 2026-09-18 — Step 1 is missing the rule label, which is a defect as per the contract.gpt-oss:20b: fail (error) 2026-09-18 — Step 3 applies the chain rule (derivative of cos(u) = -sin(u)·u') but is labeled as "trig" and combines two rules in one step. This violates the rule that each step must change only one thing and use an appropriate label.gpt-oss:20b: pass 2026-09-17deepseek-r1:70b: pass 2026-09-17
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
gemma4:26b, checked 2026-09-26 with SymPy 1.14.0.