Derivative of \( \displaystyle - \frac{\ln{\left(\cos{\left(2 x - 3 \right)} \right)}}{2} \)
Problem 2.1825 · hard Beautiful
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cos{\left(2 x - 3 \right)} \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\cos{\left(2 x - 3 \right)} \right)}}{2}\right) \]constant-multipleStart with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\cos{\left(2 x - 3 \right)} \right)}}{2} \]logarithmicApply the derivative rule for the logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(2 x - 3 \right)}}{2 \cos{\left(2 x - 3 \right)}} \]trigApply the derivative rule for the cosine function.✓ Proved
- \[ = \frac{\sin{\left(2 x - 3 \right)} \frac{d}{d x} \left(2 x - 3\right)}{2 \cos{\left(2 x - 3 \right)}} \]derivativeApply the chain rule to the inner function.✓ Proved
- \[ = \frac{\sin{\left(2 x - 3 \right)}}{\cos{\left(2 x - 3 \right)}} \]algebra algebra simplifyDifferentiate the linear term 2*x - 3. Multiply the constants together. Simplify the expression by canceling terms.✓ Proved
- \[ = \tan{\left(2 x - 3 \right)} \]simplifyUse the tangent identity.✓ Proved
Answer \( \tan{\left(2 x - 3 \right)} \)
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 - 3) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(2*x - 3) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(2*x - 3) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(2*x - 3) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(2*x - 3) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(2*x - 3) = 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 4 applies both the derivative of cosine and the chain rule but is labeled only as "derivative"; step 5 differentiates a linear term yet is labeled "algebra". These labeling errors violate the single‑rule‑per‑step rule.qwen3.6:27b-mlx: fail (error) — Step 1 is labeled 'constant-multiple' but performs no operation; it merely restates the problem. Step 2 is labeled 'logarithmic' but performs the constant-multiple rule (pulling out 1/2) and the logarithmic derivative rule simultaneously, violating the one-rule-per-step constraint.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-08 — Step 1 is labeled 'constant-multiple' but performs no operation; it merely restates the problem. Step 2 is labeled 'logarithmic' but performs the constant-multiple rule (pulling out 1/2) and the logarithmic derivative rule simultaneously, violating the one-rule-per-step constraint.gpt-oss:20b: fail (style) 2026-10-08 — Step 4 applies both the derivative of cosine and the chain rule but is labeled only as "derivative"; step 5 differentiates a linear term yet is labeled "algebra". These labeling errors violate the single‑rule‑per‑step rule.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — Step 1 is labeled 'constant-multiple' but performs no operation; it merely restates the problem. Step 2 is labeled 'logarithmic' but performs the constant-multiple rule (pulling out 1/2), not the logarithmic derivative rule. Step 3 is labeled 'trig' but applies the chain rule for the logarithm, not a trigonometric derivative rule.
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.