Derivative of \( \displaystyle - \cos{\left(2 x - 1 \right)} \)
Problem 2.875 · easy
Differentiate \( \displaystyle f(x) = - \cos{\left(2 x - 1 \right)} \).
- \[ \frac{d}{d x} \left(- \cos{\left(2 x - 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} \cos{\left(2 x - 1 \right)} \]constantPull out the constant factor -1.✓ Proved
- \[ = \sin{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right) \]trig algebraApply the derivative rule for cosine. Simplify the signs.✓ Proved
- \[ = 2 \sin{\left(2 x - 1 \right)} \]derivative algebraDifferentiate the inner function 2*x - 1. Final simplified expression.✓ Proved
Answer \( 2 \sin{\left(2 x - 1 \right)} \)
✓ 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.
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: passqwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 2 applies the constant multiple rule, but is labeled 'constant', which is not the correct label for pulling out a factor. Step 3 applies the chain rule, but is labeled 'trig', which is imprecise given the vocabulary allows 'chain'. Step 5 applies the derivative of a linear function, labeled 'derivative', which is acceptable, but the previous steps have labeling defects.gpt-oss:20b: pass 2026-09-26
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.