Derivative of \( \displaystyle - \frac{\cos{\left(4 x - 3 \right)}}{4} \)
Problem 2.578 · medium
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(4 x - 3 \right)}}{4} \).
- \[ \frac{d}{d x} \left(- \frac{\cos{\left(4 x - 3 \right)}}{4}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(4 x - 3 \right)}}{4} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\sin{\left(4 x - 3 \right)} \frac{d}{d x} \left(4 x - 3\right)}{4} \]trigApply the derivative rule for cosine.✓ Proved
- \[ = \sin{\left(4 x - 3 \right)} \]derivative simplifyDifferentiate the inner function 4*x - 3. Simplify the resulting expression.✓ Proved
Answer \( \sin{\left(4 x - 3 \right)} \)
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 both the cosine derivative rule and the chain rule in one line, but only labels it as "trig". This violates the rule that each step must change only one thing and must label the correct rule(s).qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (error) 2026-09-21 — Step 3 applies both the cosine derivative rule and the chain rule in one line, but only labels it as "trig". This violates the rule that each step must change only one thing and must label the correct rule(s).qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the constant multiple rule, chain rule (labeled as trig/derivative steps), and simplification. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: fail (style) 2026-09-21 — Step 3 incorrectly labels the application of the chain rule as "trig"; the correct label should be "chain" (or "product" if treating the product of -sin and the inner derivative).
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.