Derivative of \( \displaystyle - \ln{\left(\cos{\left(2 x - 1 \right)} \right)} \)
Problem 2.1836 · hard
Differentiate \( \displaystyle f(x) = - \ln{\left(\cos{\left(2 x - 1 \right)} \right)} \).
- \[ \frac{d}{d x} \left(- \ln{\left(\cos{\left(2 x - 1 \right)} \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} \ln{\left(\cos{\left(2 x - 1 \right)} \right)} \]constantMove the negative sign outside the derivative.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(2 x - 1 \right)}}{\cos{\left(2 x - 1 \right)}} \]logarithmicApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{\sin{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right)}{\cos{\left(2 x - 1 \right)}} \]chainApply the chain rule to the cosine function.✓ Proved
- \[ = \frac{2 \sin{\left(2 x - 1 \right)}}{\cos{\left(2 x - 1 \right)}} \]derivative algebraDifferentiate the inner linear function. Simplify the signs and constants.✓ Proved
- \[ = 2 \tan{\left(2 x - 1 \right)} \]simplifyUse the identity tan(u) = sin(u)/cos(u).✓ Proved
Answer \( 2 \tan{\left(2 x - 1 \right)} \)
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(2*x - 1) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(2*x - 1) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(2*x - 1) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(2*x - 1) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(2*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
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-08gpt-oss:20b: pass 2026-10-08gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (misleading) 2026-10-08 — Step 3 is labeled 'logarithmic' but the note explicitly states 'Apply the chain rule for the logarithm.' The label 'logarithmic' correctly identifies the derivative rule for log(u), but the note is misleading because it attributes the step to the chain rule, which is actually applied in Step 4. This conflates the derivative of the outer function with the chain rule application.
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 2026-10-08 with SymPy 1.14.0.