Derivative of \( \displaystyle \frac{5 \ln{\left(\sin{\left(3 x + 2 \right)} \right)}}{3} \)
Problem 2.1938 · hard
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\sin{\left(3 x + 2 \right)} \right)}}{3} \).
- \[ \frac{d}{d x} \frac{5 \ln{\left(\sin{\left(3 x + 2 \right)} \right)}}{3} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \ln{\left(\sin{\left(3 x + 2 \right)} \right)}}{3} \]constant-multiple algebra chainPull out the constant factor. This step was unnecessary and incorrect in the previous attempt; we will proceed directly to the chain rule. Apply the chain rule to the logarithm.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \sin{\left(3 x + 2 \right)}}{3 \sin{\left(3 x + 2 \right)}} \]chainApply the chain rule to the sine function.✓ Proved
- \[ = \frac{5 \cos{\left(3 x + 2 \right)} \frac{d}{d x} \left(3 x + 2\right)}{3 \sin{\left(3 x + 2 \right)}} \]chainApply the chain rule to the inner linear function.✓ Proved
- \[ = \frac{5 \cos{\left(3 x + 2 \right)}}{\sin{\left(3 x + 2 \right)}} \]derivative algebraDifferentiate the innermost term. Simplify the expression by multiplying the constants.✓ Proved
- \[ = 5 \cot{\left(3 x + 2 \right)} \]simplifyUse the identity cot(u) = cos(u)/sin(u).✓ Proved
Answer \( \frac{5}{\tan{\left(3 x + 2 \right)}} \)
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.
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 sin(3*x + 2) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x + 2) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x + 2) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x + 2) = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x + 2) = 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 + 2) = 0 |
| 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 rewrites log(sin(3*x+2)) as (sin(3*x+2)/sin(3*x+2))*log(sin(3*x+2)), which changes the expression and applies two algebraic manipulations at once (introducing a factor of 1 and multiplying by sin). This is not a valid single-rule application and invalidates the subsequent steps.qwen3.6:27b-mlx: fail (error) — Step 3 is mathematically incorrect; it claims that log(u) equals u/u * log(u) inside the derivative operator, which is a nonsensical algebraic manipulation that does not preserve equality in the context of differentiation rules. Furthermore, the final answer 5/tan(3*x + 2) is incorrect; the correct derivative is 5*cot(3*x + 2), and 5/tan is not equal to 5*cot.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-09 — Step 3 rewrites log(sin(3*x+2)) as (sin(3*x+2)/sin(3*x+2))*log(sin(3*x+2)), which changes the expression and applies two algebraic manipulations at once (introducing a factor of 1 and multiplying by sin). This is not a valid single-rule application and invalidates the subsequent steps.qwen3.6:27b-mlx: fail (error) 2026-10-09 — Step 3 is mathematically incorrect; it claims that log(u) equals u/u * log(u) inside the derivative operator, which is a nonsensical algebraic manipulation that does not preserve equality in the context of differentiation rules. Furthermore, the final answer 5/tan(3*x + 2) is incorrect; the correct derivative is 5*cot(3*x + 2), and 5/tan is not equal to 5*cot.qwen3.6:27b-mlx: fail (error) 2026-10-09 — Step 3 introduces a mathematically incorrect expression (multiplying by sin/sin inside the derivative) and claims it is 'algebra', which is false. Step 4 then reverts to the previous state, making Step 3 a nonsensical, erroneous detour that violates the requirement for correct mathematical logic.gpt-oss:20b: pass 2026-10-09
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 2026-10-09 with SymPy 1.14.0.