Derivative of \( \displaystyle - \frac{3 \ln{\left(\cos{\left(4 x \right)} \right)}}{4} \)
Problem 2.282 · hard Beautiful
Differentiate \( \displaystyle f(x) = - \frac{3 \ln{\left(\cos{\left(4 x \right)} \right)}}{4} \).
- \[ \frac{d}{d x} \left(- \frac{3 \ln{\left(\cos{\left(4 x \right)} \right)}}{4}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{3 \frac{d}{d x} \ln{\left(\cos{\left(4 x \right)} \right)}}{4} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{3 \frac{d}{d x} \cos{\left(4 x \right)}}{4 \cos{\left(4 x \right)}} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{3 \sin{\left(4 x \right)} \frac{d}{d x} 4 x}{4 \cos{\left(4 x \right)}} \]chainApply the chain rule to the cosine function.✓ Proved
- \[ = \frac{3 \sin{\left(4 x \right)}}{\cos{\left(4 x \right)}} \]derivative algebra algebraDifferentiate the innermost function 4*x. Simplify the constants (-3/4 * 4). Simplify the signs and the fraction.✓ Proved
- \[ = 3 \tan{\left(4 x \right)} \]simplifyUse the identity tan(u) = sin(u)/cos(u).✓ Proved
Answer \( 3 \tan{\left(4 x \right)} \)
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.
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 |
| 8 | ✓ 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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: pass — The solution correctly applies the chain rule and constant multiple rule in distinct steps. The algebraic simplifications are sound and properly labeled.
Every verdict on record (12)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule and constant multiple rule in distinct steps. The algebraic simplifications are sound and properly labeled.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: fail 2026-09-17 — The solution ignores the domain restriction that cos(4x) must be positive for log(cos(4x)) to be defined, which is a branch‑cut issue that should be noted.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.