Derivative of \( \displaystyle - 2 \ln{\left(\cos{\left(x + 2 \right)} \right)} \)
Problem 2.1524 · hard
Differentiate \( \displaystyle f(x) = - 2 \ln{\left(\cos{\left(x + 2 \right)} \right)} \).
- \[ \frac{d}{d x} \left(- 2 \ln{\left(\cos{\left(x + 2 \right)} \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - 2 \frac{d}{d x} \ln{\left(\cos{\left(x + 2 \right)} \right)} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{2 \frac{d}{d x} \cos{\left(x + 2 \right)}}{\cos{\left(x + 2 \right)}} \]logarithmic algebraApply the chain rule for the logarithm. Rewrite the fraction for clarity.✓ Proved
- \[ = \frac{2 \sin{\left(x + 2 \right)} \frac{d}{d x} \left(x + 2\right)}{\cos{\left(x + 2 \right)}} \]trigDifferentiate the cosine function using the chain rule.✓ Proved
- \[ = \frac{2 \sin{\left(x + 2 \right)}}{\cos{\left(x + 2 \right)}} \]derivative algebraDifferentiate the inner function x + 2. Simplify the signs and the expression.✓ Proved
- \[ = 2 \tan{\left(x + 2 \right)} \]trigUse the identity sin(u)/cos(u) = tan(u).✓ Proved
Answer \( 2 \tan{\left(x + 2 \right)} \)
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(x + 2) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x + 2) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x + 2) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x + 2) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x + 2) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(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 claims it applies the 'chain rule'; the label should be 'chain' or the note should reference the logarithmic derivative rule. Step 5 is labeled 'trig' but applies the chain rule to differentiate cosine; the label should be 'chain'.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-04 — Step 3 is labeled 'logarithmic' but the note claims it applies the 'chain rule'; the label should be 'chain' or the note should reference the logarithmic derivative rule. Step 5 is labeled 'trig' but applies the chain rule to differentiate cosine; the label should be 'chain'.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.