Derivative of \( \displaystyle - \frac{\cos{\left(5 x + 1 \right)}}{5} \)
Problem 2.854 · medium Video
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(5 x + 1 \right)}}{5} \).
- \[ \frac{d}{d x} \left(- \frac{\cos{\left(5 x + 1 \right)}}{5}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(5 x + 1 \right)}}{5} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\sin{\left(5 x + 1 \right)} \frac{d}{d x} \left(5 x + 1\right)}{5} \]trigApply the derivative rule for cosine.✓ Proved
- \[ = \sin{\left(5 x + 1 \right)} \]chain simplifyApply the chain rule to the inner function. Simplify the expression by multiplying the constants.✓ Proved
Answer \( \sin{\left(5 x + 1 \right)} \)
✓ 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: passqwen3.6:27b-mlx: fail (style) — Step 3 applies the chain rule (differentiating the inner function) but is labeled 'trig'. Step 4 is labeled 'chain' but only performs algebraic simplification of the derivative of the inner function already computed in the previous step. The rules are misapplied and mislabeled.
Senior review claude-sonnet-5-5, 2026-10-13: pass — Every line is correct and verified by SymPy. The labels trig, chain and simplify each name a rule that the step actually applied, and each step changes one thing.
qwen3.6:27b-mlx: dismiss — Step 3 applies the cosine derivative rule (trig) and leaves the inner derivative unevaluated as Derivative(5*x+1, x); step 4 then evaluates that inner derivative, which is part of a chain-rule step, so both labels are legitimate and SymPy verified each line.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-13 — Every line is correct and verified by SymPy. The labels trig, chain and simplify each name a rule that the step actually applied, and each step changes one thing.qwen3.6:27b-mlx: fail (style) 2026-09-27 — Step 3 applies the chain rule (differentiating the inner function) but is labeled 'trig'. Step 4 is labeled 'chain' but only performs algebraic simplification of the derivative of the inner function already computed in the previous step. The rules are misapplied and mislabeled.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 4 is labeled 'chain', but the chain rule was already applied in Step 3 (which produced the derivative of the inner function). Step 4 merely computes the derivative of the linear inner function (5*x + 1), which should be labeled 'derivative' or 'algebra'. Additionally, Step 3 combines the trig derivative and the chain rule application into one step, which violates the 'one rule per step' constraint if Step 4 is meant to be separate, or Step 4 is redundant.gpt-oss:20b: fail (style) 2026-09-26 — Step 1 lacks a rule label, and step 3 incorrectly uses the label "trig" instead of "derivative" for the derivative of cos. Both violate the rule‑labeling requirement.qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 4 is labeled 'chain', but the chain rule was already applied in Step 3 (which produced the derivative of the inner function). Step 4 merely computes the derivative of the linear inner function (5*x + 1), which is a 'derivative' step, not a chain rule step. Applying the chain rule twice or mislabeling the evaluation of the inner derivative is a defect.gpt-oss:20b: fail (style) 2026-09-26 — Step 1 lacks a rule label; it should be marked with the "derivative" rule to satisfy the labeling requirement.
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.