Derivative of \( \displaystyle \frac{\ln{\left(\cos{\left(4 x - 1 \right)} \right)}}{4} \)
Problem 2.1916 · hard
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(4 x - 1 \right)} \right)}}{4} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\cos{\left(4 x - 1 \right)} \right)}}{4} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\cos{\left(4 x - 1 \right)} \right)}}{4} \]logarithmicApply the derivative rule for the natural logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \cos{\left(4 x - 1 \right)}}{4 \cos{\left(4 x - 1 \right)}} \]chain algebraApply the chain rule to the cosine function. Combine the fraction terms.✓ Proved
- \[ = - \frac{\sin{\left(4 x - 1 \right)} \frac{d}{d x} \left(4 x - 1\right)}{4 \cos{\left(4 x - 1 \right)}} \]trigDifferentiate the cosine function.✓ Proved
- \[ = - \frac{\sin{\left(4 x - 1 \right)}}{\cos{\left(4 x - 1 \right)}} \]derivative algebra simplifyDifferentiate the inner linear function. Multiply the terms together. Cancel the common factor of 4.✓ Proved
- \[ = - \tan{\left(4 x - 1 \right)} \]simplifyUse the identity sin(u)/cos(u) = tan(u).✓ Proved
Answer \( - \tan{\left(4 x - 1 \right)} \)
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(4*x - 1) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x - 1) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x - 1) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x - 1) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x - 1) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x - 1) = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x - 1) = 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: passqwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09gpt-oss:20b: fail (style) 2026-10-09 — Step 3 applies the logarithmic derivative rule (u′/u) but is labeled "chain", which is incorrect labeling. The step changes only one rule application, but the rule name does not match the operation performed.qwen3.6:27b-mlx: fail (error) 2026-10-09 — Step 2 is labeled 'logarithmic' but fails to apply the chain rule, leaving the derivative operator on the inner function. The transition from Step 1 to Step 2 is not a valid single-step application of the logarithmic derivative rule, which requires differentiating the outer function completely.
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 2026-10-09 with SymPy 1.14.0.