Derivative of \( \displaystyle - \ln{\left(\cos{\left(x + 1 \right)} \right)} \)
Problem 2.1675 · hard
Differentiate \( \displaystyle f(x) = - \ln{\left(\cos{\left(x + 1 \right)} \right)} \).
- \[ \frac{d}{d x} \left(- \ln{\left(\cos{\left(x + 1 \right)} \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} \ln{\left(\cos{\left(x + 1 \right)} \right)} \]constant rewrite logarithmicMove the negative sign outside. Not needed here, but let's apply the chain rule directly. Apply the rule for the derivative of a logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(x + 1 \right)}}{\cos{\left(x + 1 \right)}} \]chainApply the chain rule to the inner function.✓ Proved
- \[ = \frac{\sin{\left(x + 1 \right)} \frac{d}{d x} \left(x + 1\right)}{\cos{\left(x + 1 \right)}} \]trigDifferentiate the cosine function.✓ Proved
- \[ = \frac{\sin{\left(x + 1 \right)}}{\cos{\left(x + 1 \right)}} \]derivative algebraDifferentiate the inner linear function. Simplify the signs and the fraction.✓ Proved
- \[ = \tan{\left(x + 1 \right)} \]simplifyUse the trigonometric identity for tangent.✓ Proved
Answer \( \tan{\left(x + 1 \right)} \)
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 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where cos(x + 1) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x + 1) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x + 1) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x + 1) = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(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 3 incorrectly rewrites “log(cos(x+1))” as “cos(x+1)/cos(x+1)*log(cos(x+1))”, which is algebraically incorrect and applies two unrelated transformations at once. This invalidates the subsequent differentiation steps.qwen3.6:27b-mlx: fail (error) — Step 3 introduces an unnecessary and confusing multiplication by 1 (cos/cos) which is immediately discarded in Step 4, violating the principle of changing only one thing meaningfully. Furthermore, Step 4 is labeled 'logarithmic' but performs no differentiation; the actual differentiation of the logarithm occurs in Step 5, which is labeled 'chain'. The labeling is disjointed and the intermediate step 3 is nonsensical.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-06 — Step 3 incorrectly rewrites “log(cos(x+1))” as “cos(x+1)/cos(x+1)*log(cos(x+1))”, which is algebraically incorrect and applies two unrelated transformations at once. This invalidates the subsequent differentiation steps.qwen3.6:27b-mlx: fail (error) 2026-10-06 — Step 3 introduces an unnecessary and confusing multiplication by 1 (cos/cos) which is immediately discarded in Step 4, violating the principle of changing only one thing meaningfully. Furthermore, Step 4 is labeled 'logarithmic' but performs no differentiation; the actual differentiation of the logarithm occurs in Step 5, which is labeled 'chain'. The labeling is disjointed and the intermediate step 3 is nonsensical.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — Step 3 introduces an unnecessary and incorrect algebraic manipulation (multiplying by 1 in a weird way) that is not justified by the label 'rewrite' in the context of differentiation rules, and Step 4 incorrectly labels the application of the chain rule for logarithms as 'logarithmic' while skipping the actual chain rule step that should have been explicit. More critically, Step 3 is a non-sequitur that adds terms not present in the previous step without a valid rule application, violating the 'one thing' constraint and logical flow.
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.