Derivative of \( \displaystyle \frac{\ln{\left(\cos{\left(2 x + 1 \right)} \right)}}{2} \)
Problem 2.2043 · hard
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(2 x + 1 \right)} \right)}}{2} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\cos{\left(2 x + 1 \right)} \right)}}{2} \]derivativeStart 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
- \[ = \frac{\frac{d}{d x} \cos{\left(2 x + 1 \right)}}{2 \cos{\left(2 x + 1 \right)}} \]logarithmic algebraApply the chain rule for the logarithm. Simplify the expression.✓ Proved
- \[ = - \frac{\sin{\left(2 x + 1 \right)} \frac{d}{d x} \left(2 x + 1\right)}{2 \cos{\left(2 x + 1 \right)}} \]trigApply the chain rule for the cosine function.✓ Proved
- \[ = - \frac{\sin{\left(2 x + 1 \right)}}{\cos{\left(2 x + 1 \right)}} \]derivative algebraDifferentiate the inner linear function. Simplify the constant and terms.✓ Proved
- \[ = - \tan{\left(2 x + 1 \right)} \]trigUse the identity sin(u)/cos(u) = tan(u).✓ 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.
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 |
| 8 | ✓ 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: fail (style) — Step 3 is labeled 'logarithmic' but the note says 'Apply the chain rule'; the label should be 'chain' to reflect the application of the chain rule to the logarithm. Step 5 is labeled 'trig' but the note says 'Apply the chain rule'; the label should be 'chain' to reflect the differentiation of the composite cosine function.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-11 — Step 3 is labeled 'logarithmic' but the note says 'Apply the chain rule'; the label should be 'chain' to reflect the application of the chain rule to the logarithm. Step 5 is labeled 'trig' but the note says 'Apply the chain rule'; the label should be 'chain' to reflect the differentiation of the composite cosine function.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (misleading) 2026-10-11 — Steps 3 and 5 are labeled 'logarithmic' and 'trig' respectively, but the notes explicitly state that the chain rule is being applied. The label should be 'chain' to accurately reflect the rule used, as the step involves differentiating a composite function.gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.