Derivative of \( \displaystyle \frac{2 \ln{\left(\sin{\left(3 x - 1 \right)} \right)}}{3} \)
Problem 2.1849 · hard
Differentiate \( \displaystyle f(x) = \frac{2 \ln{\left(\sin{\left(3 x - 1 \right)} \right)}}{3} \).
- \[ \frac{d}{d x} \frac{2 \ln{\left(\sin{\left(3 x - 1 \right)} \right)}}{3} \]constantStart with the derivative of the function.✓ Proved
- \[ = \frac{2 \frac{d}{d x} \ln{\left(\sin{\left(3 x - 1 \right)} \right)}}{3} \]constant-multiplePull the constant factor out of the derivative.✓ Proved
- \[ = \frac{2 \frac{d}{d x} \sin{\left(3 x - 1 \right)}}{3 \sin{\left(3 x - 1 \right)}} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{2 \cos{\left(3 x - 1 \right)} \frac{d}{d x} \left(3 x - 1\right)}{3 \sin{\left(3 x - 1 \right)}} \]chainApply the chain rule for the sine function.✓ Proved
- \[ = \frac{2 \cos{\left(3 x - 1 \right)}}{\sin{\left(3 x - 1 \right)}} \]derivative algebra algebraDifferentiate the inner linear function. Multiply the constants 2/3 and 3. Combine the terms into a single fraction.✓ Proved
- \[ = 2 \cot{\left(3 x - 1 \right)} \]simplifyUse the cotangent identity.✓ Proved
Answer \( \frac{2}{\tan{\left(3 x - 1 \right)}} \)
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.
✓ 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 sin(3*x - 1) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x - 1) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x - 1) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x - 1) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x - 1) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x - 1) = 0 cot has poles at multiples of pi |
| 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 undefined where tan(3*x - 1) = 0 |
| 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 — The solution correctly applies the chain rule and constant multiple rule in distinct steps. The final simplification to cotangent is algebraically equivalent to the stated answer 2/tan(3*x - 1).
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies the chain rule and constant multiple rule in distinct steps. The final simplification to cotangent is algebraically equivalent to the stated answer 2/tan(3*x - 1).gpt-oss:20b: pass 2026-10-08gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (style) 2026-10-08 — Step 1 is labeled 'constant' but performs no operation; it merely restates the problem, which violates the requirement that each step changes one thing. Step 8 simplifies to 2*cot(3*x - 1), but the stated answer is 2/tan(3*x - 1); while algebraically equivalent, the solution does not match the stated answer form.
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.