Derivative of \( \displaystyle \frac{\ln{\left(\sin{\left(2 x + 1 \right)} \right)}}{2} \)
Problem 2.1596 · hard
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\sin{\left(2 x + 1 \right)} \right)}}{2} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\sin{\left(2 x + 1 \right)} \right)}}{2} \]constant-multiplePull out the constant factor 1/2.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\sin{\left(2 x + 1 \right)} \right)}}{2} \]logarithmicApply the rule for the derivative of a logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \sin{\left(2 x + 1 \right)}}{2 \sin{\left(2 x + 1 \right)}} \]trigApply the chain rule to the sine function.✓ Proved
- \[ = \frac{\cos{\left(2 x + 1 \right)} \frac{d}{d x} \left(2 x + 1\right)}{2 \sin{\left(2 x + 1 \right)}} \]derivativeDifferentiate the inner linear function.✓ Proved
- \[ = \frac{\cos{\left(2 x + 1 \right)}}{\sin{\left(2 x + 1 \right)}} \]algebra simplifyMultiply the constants. Simplify the expression.✓ Proved
- \[ = \cot{\left(2 x + 1 \right)} \]simplifyUse the cotangent identity.✓ Proved
Answer \( \frac{1}{\tan{\left(2 x + 1 \right)}} \)
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 sin(2*x + 1) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x + 1) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x + 1) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x + 1) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x + 1) = 0 cot has poles at multiples of pi |
| 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 undefined where tan(2*x + 1) = 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: fail (error) — Step 3 is labeled 'trig' but applies the chain rule; the label must be 'chain'. Step 4 is labeled 'derivative' but applies the chain rule to the sine function; the label must be 'chain'. The labels do not match the rules applied.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 3 is labeled 'trig' but applies the chain rule; the label must be 'chain'. Step 4 is labeled 'derivative' but applies the chain rule to the sine function; the label must be 'chain'. The labels do not match the rules applied.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (misleading) 2026-10-05 — Step 3 is labeled 'trig' but applies the chain rule; the label should be 'chain'. Step 4 is labeled 'derivative' but applies the chain rule to the sine function; the label should be 'chain' or 'trig' depending on interpretation, but 'derivative' usually implies a basic derivative rule like d/dx(x^n). More critically, Step 2 is labeled 'logarithmic' but applies the chain rule to the log function (d/dx log(u) = u'/u); the label 'logarithmic' is acceptable if it implies the log derivative rule, but Step 3's label 'trig' for a chain rule application is incorrect. The primary defect is the mislabeling of the chain rule steps.gpt-oss:20b: fail (error) 2026-10-05 — Step 4 applies two rules at once: it differentiates the sine function (trig rule) and also differentiates the inner linear function (derivative rule). This violates the one‑rule‑per‑step requirement. Additionally, the label "derivative" is inappropriate for the chain‑rule application in that step.
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 2026-10-05 with SymPy 1.14.0.