Home›Calculus 1›Chain rule›Problem 2.203

Derivative of \( \displaystyle \frac{\ln{\left(\cos{\left(3 x \right)} \right)}}{3} \)

Problem 2.203 · hard Video

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 \( - \tan{\left(3 x \right)} \)

Top: the graph of f = log(cos(3x))/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -sin(3x)/cos(3x).

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.

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. How we verify

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'.

Senior review claude-sonnet-5-5, 2026-10-06: pass — Every step is a valid single change and its label names a rule that was actually applied. The final answer -tan(3x) is correct, since d/dx[log(cos 3x)/3] = -tan 3x.

gpt-oss:20b: dismiss — Differentiating cos(3x) to -sin(3x)·Derivative(3x) is a legitimate trig-derivative step, and evaluating Derivative(3*x) to 3 is correctly labeled 'derivative'; neither label names a rule that was not applied.
qwen3.6:27b-mlx: dismiss — Each label describes the step leaving its line: the 1→2 step is labeled constant-multiple, and the 2→3 step is labeled chain, which is correct because it turns d/dx log(u) into (1/u)·u'. Line 4's 'derivative' label is also valid because that step evaluates Derivative(3*x).

Every verdict on record (16)

claude-sonnet-5-5: pass 2026-10-06 — Every step is a valid single change and its label names a rule that was actually applied. The final answer -tan(3x) is correct, since d/dx[log(cos 3x)/3] = -tan 3x.
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 2026-09-19
deepseek-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 2026-09-19
qwen3.6:27b-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 2026-09-19
gpt-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 2026-09-18
gpt-oss:20b: pass 2026-09-18
gpt-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.