Derivative of \( \displaystyle - 5 \ln{\left(\cos{\left(x \right)} \right)} \)
Problem 2.309 · hard
Differentiate \( \displaystyle f(x) = - 5 \ln{\left(\cos{\left(x \right)} \right)} \).
- \[ \frac{d}{d x} \left(- 5 \ln{\left(\cos{\left(x \right)} \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - 5 \frac{d}{d x} \ln{\left(\cos{\left(x \right)} \right)} \]constantPull the constant factor out of the derivative.✓ Proved
- \[ = - \frac{5 \frac{d}{d x} \cos{\left(x \right)}}{\cos{\left(x \right)}} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{5 \sin{\left(x \right)}}{\cos{\left(x \right)}} \]derivative algebraDifferentiate the inner function cos(x). Simplify the signs and the fraction.✓ Proved
- \[ = 5 \tan{\left(x \right)} \]simplifyUse the trigonometric identity sin(x)/cos(x) = tan(x).✓ Proved
Answer \( 5 \tan{\left(x \right)} \)
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(x) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(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
Every verdict on record (12)
qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 2 uses the label 'constant', but the operation performed is pulling out a constant factor, which corresponds to the 'constant-multiple' rule in the provided vocabulary. The label 'constant' typically refers to the derivative of a constant being zero.gpt-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 of log(cos(x)). The original function is only defined where cos(x)>0, so its derivative is only valid there, whereas the final simplified form 5*tan(x) is defined for all x with cos(x)≠0, including points where the original function is undefined.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.