Derivative of \( \displaystyle - \frac{\cos{\left(3 x + 2 \right)}}{3} \)
Problem 2.601 · medium Video
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(3 x + 2 \right)}}{3} \).
- \[ \frac{d}{d x} \left(- \frac{\cos{\left(3 x + 2 \right)}}{3}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(3 x + 2 \right)}}{3} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\sin{\left(3 x + 2 \right)} \frac{d}{d x} \left(3 x + 2\right)}{3} \]trigApply the derivative rule for cosine.✓ Proved
- \[ = \sin{\left(3 x + 2 \right)} \]chain simplifyApply the chain rule to the inner function. Simplify the resulting expression.✓ Proved
Answer \( \sin{\left(3 x + 2 \right)} \)
✓ 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. How we verify
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 (style) — Step 1 lacks a rule label; it should be marked as "derivative".qwen3.6:27b-mlx: fail (error) — Step 3 is labeled 'trig' but applies the chain rule (differentiating the inner function), which is a mislabeling of the differentiation rule applied. Step 4 is labeled 'chain' but performs algebraic simplification of the derivative of the inner function, which is also a mislabeling.
Senior review claude-sonnet-5-5, 2026-10-10: pass — Each line was proved equivalent by SymPy and the rule labels are legitimate: constant-multiple, trig, chain, then simplify. The answer sin(3x+2) is correct, since d/dx[-cos(3x+2)/3] = sin(3x+2).
gpt-oss:20b: dismiss — Line 1 simply restates the problem and needs no rule label; SymPy proved it states the problem, so this is not a defect.qwen3.6:27b-mlx: dismiss — Step 3 applies the cosine derivative rule (trig) and leaves Derivative(3x+2) unevaluated, so 'trig' is correct. Step 4 evaluates the inner derivative of 3x+2, which the contract counts as part of a chain-rule step, so 'chain' is correct. Neither label is wrong.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-10 — Each line was proved equivalent by SymPy and the rule labels are legitimate: constant-multiple, trig, chain, then simplify. The answer sin(3x+2) is correct, since d/dx[-cos(3x+2)/3] = sin(3x+2).qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 3 is labeled 'trig' but applies the chain rule (differentiating the inner function), which is a mislabeling of the differentiation rule applied. Step 4 is labeled 'chain' but performs algebraic simplification of the derivative of the inner function, which is also a mislabeling.gpt-oss:20b: fail (style) 2026-09-26 — Step 1 lacks a rule label; it should be marked as "derivative".qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 is labeled 'trig' but performs the chain rule (differentiating the inner function), which is a mislabeling of the differentiation rule applied. Step 4 is labeled 'chain' but only performs algebraic simplification of the derivative of the inner function, which should be labeled 'derivative' or 'algebra'.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 4 is mislabeled; the step computes the derivative of the inner linear function (3*x + 2), which is a 'derivative' step, not the application of the chain rule itself (which was already invoked in Step 3 to introduce the derivative of the inner function).gpt-oss:20b: pass 2026-09-21
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.