Derivative of \( \displaystyle \frac{\ln{\left(\cos{\left(3 x + 1 \right)} \right)}}{3} \)
Problem 2.2000 · hard
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(3 x + 1 \right)} \right)}}{3} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\cos{\left(3 x + 1 \right)} \right)}}{3} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\cos{\left(3 x + 1 \right)} \right)}}{3} \]logarithmicApply the derivative rule for the natural logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \cos{\left(3 x + 1 \right)}}{3 \cos{\left(3 x + 1 \right)}} \]chain algebraApply the chain rule to the cosine function. Combine the fraction terms.✓ Proved
- \[ = - \frac{\sin{\left(3 x + 1 \right)} \frac{d}{d x} \left(3 x + 1\right)}{3 \cos{\left(3 x + 1 \right)}} \]trigDifferentiate the cosine function.✓ Proved
- \[ = - \frac{\sin{\left(3 x + 1 \right)}}{\cos{\left(3 x + 1 \right)}} \]derivative algebra simplifyDifferentiate the inner linear function. Multiply the terms together. Cancel the common factor of 3.✓ Proved
- \[ = - \tan{\left(3 x + 1 \right)} \]simplifyUse the identity sin(u)/cos(u) = tan(u).✓ 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 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(3*x + 1) = 0 |
| 9 | ✓ 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: passqwen3.6:27b-mlx: fail (error) — Step 3 is labeled 'chain' but performs the differentiation of the logarithm (applying the logarithmic rule), whereas the chain rule application for the inner cosine function occurs in Step 5. The label in Step 3 does not match the operation performed.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (error) 2026-10-10 — Step 3 is labeled 'chain' but performs the differentiation of the logarithm (applying the logarithmic rule), whereas the chain rule application for the inner cosine function occurs in Step 5. The label in Step 3 does not match the operation performed.gpt-oss:20b: fail (style) 2026-10-10 — Step 3 incorrectly labels the rule as "chain"; the derivative of log(cos(3*x+1)) is obtained by the logarithmic rule (u'/u), not a chain rule application. No other steps combine multiple rules.qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint and using valid labels from the fixed vocabulary.
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 2026-10-10 with SymPy 1.14.0.