Derivative of \( \displaystyle - \frac{\ln{\left(\cos{\left(2 x + 1 \right)} \right)}}{2} \)
Problem 2.1503 · hard Beautiful
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cos{\left(2 x + 1 \right)} \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\cos{\left(2 x + 1 \right)} \right)}}{2}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\cos{\left(2 x + 1 \right)} \right)}}{2} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(2 x + 1 \right)}}{2 \cos{\left(2 x + 1 \right)}} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{\sin{\left(2 x + 1 \right)} \frac{d}{d x} \left(2 x + 1\right)}{2 \cos{\left(2 x + 1 \right)}} \]chainApply the chain rule for the cosine function.✓ Proved
- \[ = \frac{\sin{\left(2 x + 1 \right)}}{\cos{\left(2 x + 1 \right)}} \]derivative constant-multipleDifferentiate the innermost linear function. Simplify the constant factors.✓ Proved
- \[ = \tan{\left(2 x + 1 \right)} \]algebra simplifyUse the identity sec(x) = 1/cos(x) and simplify signs. Final simplification.✓ Proved
Answer \( \tan{\left(2 x + 1 \right)} \)
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.
✓ 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 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 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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: fail (style) — Step 7 applies two transformations at once—rewriting sin/cos as tan and simplifying the double negative—yet it is labeled only as "algebra". Each step should change only one thing, so this step should be split into two separate steps.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04gpt-oss:20b: fail (style) 2026-10-04 — Step 7 applies two transformations at once—rewriting sin/cos as tan and simplifying the double negative—yet it is labeled only as "algebra". Each step should change only one thing, so this step should be split into two separate steps.qwen3.6:27b-mlx: fail (style) 2026-10-04 — Step 7 applies multiple rules simultaneously (trigonometric identity, sign simplification, and constant multiplication) under the label 'algebra', violating the one-rule-per-step constraint.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.