Derivative of \( \displaystyle - \ln{\left(\cos{\left(x - 3 \right)} \right)} \)
Problem 2.1681 · hard
Differentiate \( \displaystyle f(x) = - \ln{\left(\cos{\left(x - 3 \right)} \right)} \).
- \[ \frac{d}{d x} \left(- \ln{\left(\cos{\left(x - 3 \right)} \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} \ln{\left(\cos{\left(x - 3 \right)} \right)} \]constant rewrite logarithmicPull the constant factor out of the derivative. This step is not needed, let's use the logarithmic rule directly. Apply the derivative rule for the natural logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(x - 3 \right)}}{\cos{\left(x - 3 \right)}} \]chainApply the chain rule to the inner function.✓ Proved
- \[ = \frac{\sin{\left(x - 3 \right)}}{\cos{\left(x - 3 \right)}} \]trig algebraDifferentiate the cosine function. Simplify the signs and the fraction.✓ Proved
- \[ = \tan{\left(x - 3 \right)} \]simplifyUse the trigonometric identity for tangent.✓ Proved
Answer \( \tan{\left(x - 3 \right)} \)
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 |
| 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 - 3) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x - 3) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x - 3) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(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: fail (error) — Step 3 introduces a nonsensical term (cos(x-3)/cos(x-3)) that is not present in the previous step, violating the requirement that steps be algebraically equal. Step 4 then reverts to the expression from Step 2, making Step 3 a logical non-sequitur rather than a valid transformation.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — Step 3 introduces a nonsensical term (cos(x-3)/cos(x-3)) that is not present in the previous step, violating the requirement that steps be algebraically equal. Step 4 then reverts to the expression from Step 2, making Step 3 a logical non-sequitur rather than a valid transformation.gpt-oss:20b: fail (style) 2026-10-06 — Step 5 is labeled "chain" but the rule applied is the logarithmic derivative rule (u'/u). The labeling is incorrect, though the algebraic manipulation is fine.qwen3.6:27b-mlx: fail (error) 2026-10-06 — Step 3 introduces an unnecessary and confusing rewrite (multiplying by 1) that is immediately reversed in Step 4, violating the principle that each step should change one thing meaningfully. Furthermore, Step 4 is labeled 'logarithmic' but performs no algebraic change, merely restating the expression from Step 2, making the label and step redundant and logically disjointed.
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.