Derivative of \( \displaystyle - 5 \cos{\left(x - 3 \right)} \)
Problem 2.688 · easy
Differentiate \( \displaystyle f(x) = - 5 \cos{\left(x - 3 \right)} \).
- \[ \frac{d}{d x} \left(- 5 \cos{\left(x - 3 \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - 5 \frac{d}{d x} \cos{\left(x - 3 \right)} \]constantPull out the constant factor.✓ Proved
- \[ = 5 \sin{\left(x - 3 \right)} \frac{d}{d x} \left(x - 3\right) \]trigApply the derivative rule for cosine.✓ Proved
- \[ = 5 \sin{\left(x - 3 \right)} \]chain algebraApply the chain rule to the inner function. Simplify the signs.✓ Proved
Answer \( 5 \sin{\left(x - 3 \right)} \)
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: passqwen3.6:27b-mlx: fail (style) — Step 4 is labeled 'chain', but the chain rule was already applied in Step 3 (which produced the Derivative(x - 3, x) term). Step 4 merely evaluates the derivative of the inner function, which should be labeled 'derivative' or 'algebra', not 'chain'.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 4 is labeled 'chain', but the chain rule was already applied in Step 3 (which produced the Derivative(x - 3, x) term). Step 4 merely evaluates the derivative of the inner function, which should be labeled 'derivative' or 'algebra', not 'chain'.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (error) 2026-09-21 — Step 3 applies two rules at once (the derivative of cosine and the chain rule) but is labeled only as "trig". It should be split into separate steps or labeled appropriately. Step 1 also lacks a proper label; it should be "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.