Derivative of \( \displaystyle - \frac{\ln{\left(\cos{\left(5 x + 2 \right)} \right)}}{5} \)
Problem 2.1576 · hard Beautiful
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cos{\left(5 x + 2 \right)} \right)}}{5} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\cos{\left(5 x + 2 \right)} \right)}}{5}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\cos{\left(5 x + 2 \right)} \right)}}{5} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(5 x + 2 \right)}}{5 \cos{\left(5 x + 2 \right)}} \]logarithmicApply the chain rule for the natural logarithm.✓ Proved
- \[ = \frac{\sin{\left(5 x + 2 \right)} \frac{d}{d x} \left(5 x + 2\right)}{5 \cos{\left(5 x + 2 \right)}} \]trigDifferentiate the cosine function.✓ Proved
- \[ = \frac{\sin{\left(5 x + 2 \right)}}{\cos{\left(5 x + 2 \right)}} \]derivative algebraDifferentiate the linear expression inside. Simplify the constants and signs.✓ Proved
- \[ = \tan{\left(5 x + 2 \right)} \]simplifyUse the identity sin(u)/cos(u) = tan(u) and simplify the 5s.✓ Proved
Answer \( \tan{\left(5 x + 2 \right)} \)
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(5*x + 2) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x + 2) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x + 2) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x + 2) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x + 2) = 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: fail (style) — Step 3 is labeled 'logarithmic' but the note describes the chain rule; the step actually applies the chain rule to the logarithm, so 'chain' is the more appropriate label for the structural change, or 'logarithmic' if referring to the derivative of log, but the note is contradictory. More importantly, Step 7 combines simplifying constants (algebra) and trigonometric identity (trig/simplify) into one step, violating the 'one change per step' rule.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-05 — Step 3 is labeled 'logarithmic' but the note describes the chain rule; the step actually applies the chain rule to the logarithm, so 'chain' is the more appropriate label for the structural change, or 'logarithmic' if referring to the derivative of log, but the note is contradictory. More importantly, Step 7 combines simplifying constants (algebra) and trigonometric identity (trig/simplify) into one step, violating the 'one change per step' rule.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (style) 2026-10-05 — Step 3 is labeled 'logarithmic' but the note explicitly states 'Apply the chain rule', and the step performs the chain rule application for the logarithm; the label should be 'chain'. Step 7 is labeled 'simplify' but the note describes using the trigonometric identity sin/cos = tan, so the label should be 'trig'.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.