Derivative of \( \displaystyle \cos{\left(x - 1 \right)} \)
Problem 2.392 · easy Mental math Video
Differentiate \( \displaystyle f(x) = \cos{\left(x - 1 \right)} \).
- \[ \frac{d}{d x} \cos{\left(x - 1 \right)} \]Start with the derivative of the function.✓ Proved
- \[ = - \sin{\left(x - 1 \right)} \]trigThe derivative of cos(u) is -sin(u).✓ Proved
Answer \( - \sin{\left(x - 1 \right)} \)
✓ 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 | line 1 is the problem as stated |
| 2 | ✓ 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 is labeled with an unlabelled rule; it should be named "derivative" to indicate the application of the derivative rule to the entire function.qwen3.6:27b-mlx: fail (error) — Step 2 applies the chain rule implicitly by differentiating cos(x-1) to -sin(x-1) without multiplying by the derivative of the inner function (which is 1). While the result is correct, the label 'trig' only accounts for the outer derivative, failing to acknowledge the chain rule application required for composite functions. A step that differentiates a composite function must either explicitly show the chain rule multiplication or be labeled 'chain'.
Senior review claude-sonnet-5-5, 2026-10-08: pass — d/dx cos(x-1) = -sin(x-1) is correct, SymPy proved every line, and the 'trig' label is acceptable. The panel objections are about labelling preferences, not defects.
gpt-oss:20b: dismiss — Line 1 only restates the problem, and the contract does not require a rule label there; SymPy proved it equals the stated problem.qwen3.6:27b-mlx: dismiss — The inner function x-1 has derivative 1, and evaluating the derivative of a simple inner function is part of the step. 'trig' is a legitimate label for the cos-to-negative-sin step, and when two labels fit, either is acceptable.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-08 — d/dx cos(x-1) = -sin(x-1) is correct, SymPy proved every line, and the 'trig' label is acceptable. The panel objections are about labelling preferences, not defects.qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 2 applies the chain rule implicitly by differentiating cos(x-1) to -sin(x-1) without multiplying by the derivative of the inner function (which is 1). While the result is correct, the label 'trig' only accounts for the outer derivative, failing to acknowledge the chain rule application required for composite functions. A step that differentiates a composite function must either explicitly show the chain rule multiplication or be labeled 'chain'.gpt-oss:20b: fail (style) 2026-09-21 — Step 1 is labeled with an unlabelled rule; it should be named "derivative" to indicate the application of the derivative rule to the entire function.qwen3.6:27b-mlx: fail (error) 2026-09-20 — The step from line 1 to line 2 applies the chain rule (differentiating the inner function x-1) but is labeled only as 'trig'. This violates the single-rule-per-step constraint and mislabels the operation, as 'trig' alone does not account for the derivative of the argument.gpt-oss:20b: fail (style) 2026-09-20 — Step 1 is unlabeled; it should be marked with the rule "derivative". The subsequent step correctly applies the trig rule, but the missing label makes the solution less clear.qwen3.6:27b-mlx: fail (error) 2026-09-20 — The step from line 1 to line 2 applies the chain rule (differentiating the inner function x-1) but is labeled only as 'trig'. This violates the single-rule-per-step constraint and mislabels the application of the chain rule.gpt-oss:20b: fail (style) 2026-09-20 — Step 1 is labeled "unlabelled" even though it applies the derivative rule; it should be marked as a "derivative" step. No other defects are present.
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.