Derivative of \( \displaystyle \frac{5 \ln{\left(\sin{\left(3 x + 1 \right)} \right)}}{3} \)
Problem 2.1850 · hard
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\sin{\left(3 x + 1 \right)} \right)}}{3} \).
- \[ \frac{d}{d x} \frac{5 \ln{\left(\sin{\left(3 x + 1 \right)} \right)}}{3} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \ln{\left(\sin{\left(3 x + 1 \right)} \right)}}{3} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \sin{\left(3 x + 1 \right)}}{3 \sin{\left(3 x + 1 \right)}} \]chainApply the chain rule to the sine function.✓ Proved
- \[ = \frac{5 \cos{\left(3 x + 1 \right)} \frac{d}{d x} \left(3 x + 1\right)}{3 \sin{\left(3 x + 1 \right)}} \]chainApply the chain rule to the inner linear function.✓ Proved
- \[ = \frac{5 \cos{\left(3 x + 1 \right)}}{\sin{\left(3 x + 1 \right)}} \]derivative algebraDifferentiate the innermost term. Multiply the constants and simplify the fraction.✓ Proved
- \[ = 5 \cot{\left(3 x + 1 \right)} \]simplifyUse the cotangent identity.✓ Proved
Answer \( \frac{5}{\tan{\left(3 x + 1 \right)}} \)
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 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: fail (style) — Step 1 is labeled 'constant-multiple' but does not perform the pull-out operation; it merely restates the function with the constant factored. The actual pull-out occurs in Step 2, which is mislabeled as 'chain' instead of 'constant-multiple'.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-08 — Step 1 is labeled 'constant-multiple' but does not perform the pull-out operation; it merely restates the function with the constant factored. The actual pull-out occurs in Step 2, which is mislabeled as 'chain' instead of 'constant-multiple'.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-multiple' but does not actually pull out the constant; it merely rewrites the function with the constant factored, which is a 'rewrite' or 'algebra' step. The actual application of the constant multiple rule occurs in Step 2.
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.