Derivative of \( \displaystyle \frac{5 \sin{\left(3 x + 2 \right)}}{3} \)
Problem 2.853 · medium
Differentiate \( \displaystyle f(x) = \frac{5 \sin{\left(3 x + 2 \right)}}{3} \).
- \[ \frac{d}{d x} \frac{5 \sin{\left(3 x + 2 \right)}}{3} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \sin{\left(3 x + 2 \right)}}{3} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{5 \cos{\left(3 x + 2 \right)} \frac{d}{d x} \left(3 x + 2\right)}{3} \]trigApply the derivative rule for sine.✓ Proved
- \[ = 5 \cos{\left(3 x + 2 \right)} \]derivative algebraDifferentiate the inner function. Simplify the expression by multiplying the constants.✓ Proved
Answer \( 5 \cos{\left(3 x + 2 \right)} \)
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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 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 applies two rules at once: the sine derivative (trig) and the chain rule for the inner function. It should be split into separate steps and labeled appropriately (e.g., "trig" for the sine derivative and "chain" for differentiating the inner function).qwen3.6:27b-mlx: pass — The solution correctly applies the constant multiple rule, chain rule, and derivative of sine in distinct steps. All labels are appropriate and the logic is sound.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the constant multiple rule, chain rule, and derivative of sine in distinct steps. All labels are appropriate and the logic is sound.gpt-oss:20b: fail (error) 2026-09-26 — Step 3 applies two rules at once: the sine derivative (trig) and the chain rule for the inner function. It should be split into separate steps and labeled appropriately (e.g., "trig" for the sine derivative and "chain" for differentiating the inner function).qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the constant multiple rule, chain rule (labeled as trig for the outer derivative), and basic derivative rules in separate steps. The algebraic simplification is correct.gpt-oss:20b: fail (style) 2026-09-26 — Step 3 applies the chain rule (derivative of sin(3*x+2) = cos(3*x+2)*Derivative(3*x+2,x)), but it is labeled as "trig". The correct label from the allowed vocabulary is "chain". No algebraic or other rule is combined in that step, so the defect is only in the labeling.
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 2026-09-26 with SymPy 1.14.0.