Derivative of \( \displaystyle \frac{\ln{\left(\cos{\left(3 x - 1 \right)} \right)}}{3} \)
Problem 2.1518 · hard
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(3 x - 1 \right)} \right)}}{3} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\cos{\left(3 x - 1 \right)} \right)}}{3} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\cos{\left(3 x - 1 \right)} \right)}}{3} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\frac{d}{d x} \cos{\left(3 x - 1 \right)}}{3 \cos{\left(3 x - 1 \right)}} \]logarithmicApply the derivative rule for the natural logarithm.✓ Proved
- \[ = - \frac{\sin{\left(3 x - 1 \right)} \frac{d}{d x} \left(3 x - 1\right)}{3 \cos{\left(3 x - 1 \right)}} \]chainApply the chain rule to the cosine function.✓ Proved
- \[ = - \frac{\sin{\left(3 x - 1 \right)}}{\cos{\left(3 x - 1 \right)}} \]trig derivativeDifferentiate the sine function. Differentiate the linear inner function.✓ Proved
- \[ = - \tan{\left(3 x - 1 \right)} \]simplifySimplify the expression using the tangent identity.✓ Proved
Answer \( - \tan{\left(3 x - 1 \right)} \)
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(3*x - 1) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(3*x - 1) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(3*x - 1) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(3*x - 1) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(3*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: fail (style) — Step 5 incorrectly labels the differentiation of the linear inner function as "trig" and step 6 incorrectly labels the simplification of the constant factor as "derivative". These mislabelings violate the rule that each step must name the exact rule applied.qwen3.6:27b-mlx: fail (style) — Step 5 is labeled 'trig' but performs the differentiation of the inner linear function (3x-1), which should be labeled 'derivative'. Step 6 is labeled 'derivative' but performs algebraic simplification (canceling the 3s), which should be labeled 'simplify' or 'algebra'. The labels are swapped.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-04 — Step 5 is labeled 'trig' but performs the differentiation of the inner linear function (3x-1), which should be labeled 'derivative'. Step 6 is labeled 'derivative' but performs algebraic simplification (canceling the 3s), which should be labeled 'simplify' or 'algebra'. The labels are swapped.gpt-oss:20b: fail (style) 2026-10-04 — Step 5 incorrectly labels the differentiation of the linear inner function as "trig" and step 6 incorrectly labels the simplification of the constant factor as "derivative". These mislabelings violate the rule that each step must name the exact rule applied.qwen3.6:27b-mlx: fail (misleading) 2026-10-04 — Step 5 is labeled 'trig' but performs the differentiation of the linear inner function (3*x - 1), which is a 'derivative' step. Step 6 is labeled 'derivative' but performs algebraic simplification (canceling the 1/3 and 3), which should be 'simplify' or 'algebra'. The labels are swapped and incorrect.gpt-oss:20b: fail (style) 2026-10-04 — Steps 5 and 6 use incorrect rule labels. Step 5 differentiates the linear inner function (3 x − 1)′=3, which is a derivative operation, not a trig rule. Step 6 performs algebraic simplification (combining constants and fractions), which should be labeled "simplify" or "algebra", not "derivative". These mislabelings violate the one‑rule‑per‑step 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 2026-10-04 with SymPy 1.14.0.