Home›Calculus 1›Chain rule›Problem 2.190

Derivative of \( \displaystyle \frac{\ln{\left(\cos{\left(2 x \right)} \right)}}{2} \)

Problem 2.190 · hard Video

Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(2 x \right)} \right)}}{2} \).

\[ \frac{d}{d x} \frac{\ln{\left(\cos{\left(2 x \right)} \right)}}{2} \]
constant-multipleApply the constant multiple rule.✓ Proved
\[ = \frac{\frac{d}{d x} \ln{\left(\cos{\left(2 x \right)} \right)}}{2} \]
chainApply the chain rule to the logarithm.✓ Proved
\[ = \frac{\frac{d}{d x} \cos{\left(2 x \right)}}{2 \cos{\left(2 x \right)}} \]
trigDifferentiate the cosine function.✓ Proved
\[ = - \frac{\sin{\left(2 x \right)}}{\cos{\left(2 x \right)}} \]
derivative algebra algebra simplifyThe derivative of cos(2*x) is -sin(2*x) * 2. Rearrange the multiplication. Combine the fraction terms. Cancel the common factor of 2.✓ Proved
\[ = - \tan{\left(2 x \right)} \]
simplifyUse the identity sin(u)/cos(u) = tan(u).✓ Proved

Answer \( - \tan{\left(2 x \right)} \)

Top: the graph of f = log(cos(2x))/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -sin(2x)/cos(2x).

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. A reviewer's objection to the explanation stands; see the receipt. 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(2*x) = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(2*x) = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(2*x) = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(2*x) = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(2*x) = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(2*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: pass
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-06: fail — The mathematics is correct and every line is proved. Two rule labels are wrong, though. Step 1→2 pulls out the constant 1/2 but is labeled 'chain'. Step 2→3 applies the chain rule to the logarithm but is labeled 'trig'. The labels for lines 1–3 appear to be shifted by one step, which would teach the wrong rule names.

unspecified panel model (both passed and objected): dismiss — No specific defect is stated, and SymPy proved every line equivalent with the final answer matching -tan(2*x), so there is no mathematical error to uphold.

Every verdict on record (16)

claude-sonnet-5-5: fail (misleading) 2026-10-06 — The mathematics is correct and every line is proved. Two rule labels are wrong, though. Step 1→2 pulls out the constant 1/2 but is labeled 'chain'. Step 2→3 applies the chain rule to the logarithm but is labeled 'trig'. The labels for lines 1–3 appear to be shifted by one step, which would teach the wrong rule names.
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 2 is labeled 'chain' but performs no differentiation; it merely restates the expression from Step 1. The chain rule is actually applied in Step 3, which is incorrectly labeled 'trig'. Additionally, Step 3 applies the chain rule to the outer log function, but the label 'trig' does not describe this operation.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 2 is labeled 'chain' but performs no differentiation; it merely prepares the expression for the chain rule application in Step 3. The label should be 'algebra' or 'rewrite' to reflect that no derivative was computed in this step.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (style) 2026-09-19 — Step 3 incorrectly labels the rule as "trig"; the operation performed is the chain rule applied to the logarithm, not a trig derivative.
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (style) 2026-09-19 — Step 2 incorrectly labels the application of the constant‑multiple rule as "chain"; it merely rewrites the expression. Step 3 applies the logarithmic derivative rule but is labeled "trig", which misidentifies the rule used. These labeling errors violate the contract that each step must name the exact rule applied.
qwen3.6:27b-mlx: pass 2026-09-18
deepseek-r1:70b: pass 2026-09-18
gpt-oss:20b: fail (style) 2026-09-18 — Step 3 is labeled "trig" but only applies the chain rule to the logarithm; the derivative of the cosine is not yet taken. The correct label for this step is "chain".
gpt-oss:20b: fail 2026-09-17 — Step 2 incorrectly labels the rule as "chain" when it is still the constant‑multiple rule, and step 3’s note misleads by saying it differentiates the cosine instead of applying the chain rule to the logarithm. These mislabels could confuse a student.
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.