Derivative of \( \displaystyle - \frac{\ln{\left(\cos{\left(4 x + 1 \right)} \right)}}{4} \)
Problem 2.1687 · hard Beautiful
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cos{\left(4 x + 1 \right)} \right)}}{4} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\cos{\left(4 x + 1 \right)} \right)}}{4}\right) \]constant-multipleStart with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\cos{\left(4 x + 1 \right)} \right)}}{4} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(4 x + 1 \right)}}{4 \cos{\left(4 x + 1 \right)}} \]logarithmicApply the chain rule for the logarithm.✓ 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 constant algebra simplifyDifferentiate the inner linear function. The derivative of the constant 1 is 0, but here we differentiate the term 4*x+1 which is 4. Multiply the terms together. Simplify the expression using the tangent identity.✓ Proved
- \[ = \tan{\left(4 x + 1 \right)} \]simplifyFinal simplified form.✓ 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: fail (error) — Step 6 incorrectly applies a constant rule and introduces an unnecessary factor of 1, changing the expression in a way that is not justified by a single allowed rule. The derivative of 4*x+1 is 4, not 4*1, and the step should not multiply by 1. This violates the rule‑granularity requirement.qwen3.6:27b-mlx: fail (style) — Step 6 applies the rule 'constant' but performs no operation on a constant (the derivative of 1 is not taken; the 1 is just appended). Step 3's note incorrectly claims the 'chain rule' is applied, whereas the label 'logarithmic' is used for the derivative of log(u); the chain rule application is implicit in the structure but the note is misleading about what the specific step does versus the label.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-06 — Step 6 incorrectly applies a constant rule and introduces an unnecessary factor of 1, changing the expression in a way that is not justified by a single allowed rule. The derivative of 4*x+1 is 4, not 4*1, and the step should not multiply by 1. This violates the rule‑granularity requirement.qwen3.6:27b-mlx: fail (style) 2026-10-06 — Step 6 applies the rule 'constant' but performs no operation on a constant (the derivative of 1 is not taken; the 1 is just appended). Step 3's note incorrectly claims the 'chain rule' is applied, whereas the label 'logarithmic' is used for the derivative of log(u); the chain rule application is implicit in the structure but the note is misleading about what the specific step does versus the label.gpt-oss:20b: fail (misleading) 2026-10-06 — Step 6 contains a misleading note: it claims the derivative of the constant 1 is 0, yet still multiplies by 1. The step itself is harmless, but the explanation is incorrect and could confuse a learner.qwen3.6:27b-mlx: fail (error) 2026-10-06 — Step 6 is mathematically incorrect; it multiplies by 1 and claims the derivative of the constant 1 is 0, but the previous step already correctly differentiated 4*x+1 to 4. This step introduces a nonsensical operation and a false note.
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 2026-10-06 with SymPy 1.14.0.