Derivative of \( \displaystyle - \frac{\ln{\left(\cos{\left(5 x - 3 \right)} \right)}}{5} \)
Problem 2.1861 · hard Beautiful
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cos{\left(5 x - 3 \right)} \right)}}{5} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\cos{\left(5 x - 3 \right)} \right)}}{5}\right) \]constant-multipleStart with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\cos{\left(5 x - 3 \right)} \right)}}{5} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(5 x - 3 \right)}}{5 \cos{\left(5 x - 3 \right)}} \]logarithmicApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{\sin{\left(5 x - 3 \right)} \frac{d}{d x} \left(5 x - 3\right)}{5 \cos{\left(5 x - 3 \right)}} \]trigApply the chain rule for the cosine function.✓ Proved
- \[ = \frac{\sin{\left(5 x - 3 \right)}}{\cos{\left(5 x - 3 \right)}} \]derivative constant-multiple algebraDifferentiate the inner linear function. Simplify the constant factors. Multiply the terms together.✓ Proved
- \[ = \tan{\left(5 x - 3 \right)} \]simplifyUse the trigonometric identity tan(u) = sin(u)/cos(u).✓ Proved
Answer \( \tan{\left(5 x - 3 \right)} \)
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(5*x - 3) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x - 3) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x - 3) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x - 3) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x - 3) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x - 3) = 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 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — Step 3 is labeled 'logarithmic' but applies the chain rule to the outer logarithm; the label should be 'chain'. Step 4 is labeled 'trig' but applies the chain rule to the cosine function; the label should be 'chain'. The notes explicitly claim 'Apply the chain rule' while using incorrect labels, which is a defect in labeling.
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 2026-10-08 with SymPy 1.14.0.