Derivative of \( \displaystyle - \cos{\left(3 x + 2 \right)} \)
Problem 2.1007 · easy
Differentiate \( \displaystyle f(x) = - \cos{\left(3 x + 2 \right)} \).
- \[ \frac{d}{d x} \left(- \cos{\left(3 x + 2 \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} \cos{\left(3 x + 2 \right)} \]constantPull the constant -1 out of the derivative.✓ Proved
- \[ = \sin{\left(3 x + 2 \right)} \frac{d}{d x} \left(3 x + 2\right) \]trig algebraApply the derivative rule for cosine. Simplify the signs.✓ Proved
- \[ = \left(\frac{d}{d x} 2 + \frac{d}{d x} 3 x\right) \sin{\left(3 x + 2 \right)} \]sumApply the sum rule to the inner function.✓ Proved
- \[ = \left(\frac{d}{d x} 2 + 3\right) \sin{\left(3 x + 2 \right)} \]powerDifferentiate 3*x.✓ Proved
- \[ = 3 \sin{\left(3 x + 2 \right)} \]constant algebraThe derivative of a constant is 0. Simplify the final expression.✓ Proved
Answer \( 3 \sin{\left(3 x + 2 \right)} \)
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 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ 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 both the cosine derivative (trig) and the chain rule in one move, violating the one‑rule‑per‑step rule. Step 5 applies the sum rule to the inner function but then incorrectly labels the differentiation of 3*x as a power rule; it should be a constant‑multiple rule. These multi‑rule steps make the solution invalid.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: fail (error) 2026-09-27 — Step 3 applies both the cosine derivative (trig) and the chain rule in one move, violating the one‑rule‑per‑step rule. Step 5 applies the sum rule to the inner function but then incorrectly labels the differentiation of 3*x as a power rule; it should be a constant‑multiple rule. These multi‑rule steps make the solution invalid.qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: fail (style) 2026-09-27 — Step 6 incorrectly labels the differentiation of 3*x as "power"; it should be a constant‑multiple (or product) rule. The algebraic manipulation itself is fine, but the rule name is misleading.
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 2026-09-27 with SymPy 1.14.0.