Derivative of \( \displaystyle - 2 \ln{\left(\cos{\left(x - 1 \right)} \right)} \)
Problem 2.1898 · hard
Differentiate \( \displaystyle f(x) = - 2 \ln{\left(\cos{\left(x - 1 \right)} \right)} \).
- \[ \frac{d}{d x} \left(- 2 \ln{\left(\cos{\left(x - 1 \right)} \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - 2 \frac{d}{d x} \ln{\left(\cos{\left(x - 1 \right)} \right)} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{2 \frac{d}{d x} \cos{\left(x - 1 \right)}}{\cos{\left(x - 1 \right)}} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{2 \sin{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right)}{\cos{\left(x - 1 \right)}} \]trigDifferentiate the cosine function.✓ Proved
- \[ = \frac{2 \sin{\left(x - 1 \right)}}{\cos{\left(x - 1 \right)}} \]derivative algebraDifferentiate the inner function x - 1. Simplify the signs and the fraction.✓ Proved
- \[ = 2 \tan{\left(x - 1 \right)} \]simplifyUse the identity sin(u)/cos(u) = tan(u).✓ Proved
Answer \( 2 \tan{\left(x - 1 \right)} \)
Mind the domain. The answer is also defined at points where f(x) is not. Substituting there gives a number that is not a slope of f.
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 log is undefined for non-positive arguments |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where cos(x - 1) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x - 1) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x - 1) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x - 1) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x - 1) = 0 tan has poles at odd multiples of pi/2 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 |
| 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 — The solution correctly applies the constant multiple rule, chain rule, and trigonometric derivatives in a step-by-step manner. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies the constant multiple rule, chain rule, and trigonometric derivatives in a step-by-step manner. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09gpt-oss:20b: fail (error) 2026-10-09 — Step 4 applies two rules at once: it differentiates the cosine (trig rule) and the inner function x‑1 (derivative rule) in a single line, violating the one‑rule‑per‑step requirement.
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 2026-10-09 with SymPy 1.14.0.