Derivative of \( \displaystyle \frac{\ln{\left(\cos{\left(3 x \right)} \right)}}{3} \)
Problem 2.203 · hard
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(3 x \right)} \right)}}{3} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\cos{\left(3 x \right)} \right)}}{3} \]constant-multipleApply the constant multiple rule.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\cos{\left(3 x \right)} \right)}}{3} \]chainApply the chain rule to the natural logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \cos{\left(3 x \right)}}{3 \cos{\left(3 x \right)}} \]trigDifferentiate the cosine function.✓ Proved
- \[ = - \frac{\sin{\left(3 x \right)} \frac{d}{d x} 3 x}{3 \cos{\left(3 x \right)}} \]derivativeApply the chain rule to the inner function 3*x.✓ Proved
- \[ = - \frac{\sin{\left(3 x \right)}}{\cos{\left(3 x \right)}} \]algebra simplify trigMultiply the terms together. Cancel the common factor of 3. Use the definition of the tangent function.✓ Proved
- \[ = - \tan{\left(3 x \right)} \]simplifyFinal simplified result.✓ Proved
Answer \( - \tan{\left(3 x \right)} \)
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) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(3*x) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(3*x) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(3*x) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(3*x) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(3*x) = 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 3 incorrectly labels the differentiation of cos(3*x) as "trig"; it should be "derivative" (or "chain" if the inner 3*x is considered). Step 4 labels the application of the chain rule to 3*x as "derivative"; it should be "chain". These labeling errors do not affect the correctness of the result but misrepresent the applied rules.deepseek-r1:70b: fail (error) — Step 4 applies both the chain rule and the derivative of the inner function at once, which violates the contract.qwen3.6:27b-mlx: fail (style) — Step 2 is labeled 'chain' but performs no differentiation; it merely rewrites the expression by pulling out the constant factor, which should be labeled 'constant-multiple' or 'algebra'. Step 4 is labeled 'derivative' but applies the chain rule to the inner function, which should be labeled 'chain'.
Every verdict on record (15)
qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 2 is labeled 'chain' but performs no differentiation; it merely rewrites the expression by pulling out the constant factor, which should be labeled 'constant-multiple' or 'algebra'. Step 4 is labeled 'derivative' but applies the chain rule to the inner function, which should be labeled 'chain'.gpt-oss:20b: fail (style) 2026-09-20 — Step 3 incorrectly labels the differentiation of cos(3*x) as "trig"; it should be "derivative" (or "chain" if the inner 3*x is considered). Step 4 labels the application of the chain rule to 3*x as "derivative"; it should be "chain". These labeling errors do not affect the correctness of the result but misrepresent the applied rules.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 2 is labeled 'chain' but performs no differentiation; it merely restates the constant multiple rule applied in Step 1. The actual application of the chain rule for the logarithm occurs in Step 3, which is mislabeled as 'trig'.gpt-oss:20b: fail (style) 2026-09-20 — Step 3 incorrectly labels the derivative of log(cos(3*x)) as "trig"; it should be "logarithmic" (or "chain"). Additionally, step 4 applies both the derivative of cosine and the chain rule for the inner 3*x but labels only "derivative", which is a naming mismatch.qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: fail (error) 2026-09-19 — Step 4 applies both the chain rule and the derivative of the inner function at once, which violates the contract.gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 4 is labeled 'derivative' but applies the chain rule to cos(3*x); the label should be 'chain'. Step 5 is labeled 'algebra' but performs the differentiation of 3*x (derivative rule) and simplification simultaneously, violating the one-rule-per-step constraint.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (style) 2026-09-19 — Step 3 incorrectly labels the application of the chain rule as "trig". The derivative of log(cos(3*x)) uses the chain rule, not a trigonometric rule.qwen3.6:27b-mlx: fail (error) 2026-09-18 — Step 2 is labeled 'chain' but performs no differentiation; it merely rewrites the expression by pulling out the constant factor, which should be labeled 'constant-multiple' or 'algebra'. Step 4 is labeled 'derivative' but applies the chain rule to the inner function 3*x, which should be labeled 'chain'.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: fail 2026-09-17 — Step 3 incorrectly labels the application of the derivative of a logarithm as a chain rule, and step 4 mislabels the chain rule for the cosine as a generic derivative. These mislabels could mislead a student about which rule is actually being applied.deepseek-r1:70b: pass 2026-09-17
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 2026-09-26 with SymPy 1.14.0.