Derivative of \( \displaystyle - \frac{\ln{\left(\cos{\left(4 x \right)} \right)}}{2} \)
Problem 2.666 · hard Video
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cos{\left(4 x \right)} \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\cos{\left(4 x \right)} \right)}}{2}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\cos{\left(4 x \right)} \right)}}{2} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(4 x \right)}}{2 \cos{\left(4 x \right)}} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{\sin{\left(4 x \right)} \frac{d}{d x} 4 x}{2 \cos{\left(4 x \right)}} \]chainApply the chain rule to the cosine function.✓ Proved
- \[ = \frac{2 \sin{\left(4 x \right)}}{\cos{\left(4 x \right)}} \]derivative algebraDifferentiate the innermost function. Simplify the constants and signs.✓ Proved
- \[ = 2 \tan{\left(4 x \right)} \]simplifyUse the identity sin(x)/cos(x) = tan(x).✓ Proved
Answer \( 2 \tan{\left(4 x \right)} \)
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.
✓ 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(4*x) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*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 (error) — Step 4 applies two rules at once: it uses the chain rule on cos(4*x) and leaves the derivative of 4*x unevaluated, violating the one‑rule‑per‑step rule.qwen3.6:27b-mlx: pass — The solution correctly applies the constant multiple rule, chain rule, and basic derivative rules in a step-by-step manner. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Senior review claude-sonnet-5-5, 2026-10-11: pass — The derivation is correct. -1/2 * (1/cos)(-sin)(4) = 2 tan(4x), and every step is labeled with a rule it legitimately applied. SymPy proved each line, and the final line matches the stated answer.
gpt-oss:20b: dismiss — Step 4 applies only the chain rule to cos(4x), replacing its derivative with -sin(4x) times Derivative(4*x, x), which is left unevaluated; that is a single change with the correct label.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-11 — The derivation is correct. -1/2 * (1/cos)(-sin)(4) = 2 tan(4x), and every step is labeled with a rule it legitimately applied. SymPy proved each line, and the final line matches the stated answer.qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the constant multiple rule, chain rule, and basic derivative rules in a step-by-step manner. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-09-26 — Step 4 applies two rules at once: it uses the chain rule on cos(4*x) and leaves the derivative of 4*x unevaluated, violating the one‑rule‑per‑step rule.qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the constant multiple rule, chain rule, and basic derivatives in a step-by-step manner. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21
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.