Derivative of \( \displaystyle \cos{\left(x + 2 \right)} \)
Problem 2.433 · easy Mental math Video
Differentiate \( \displaystyle f(x) = \cos{\left(x + 2 \right)} \).
- \[ \frac{d}{d x} \cos{\left(x + 2 \right)} \]Start with the derivative of the function.✓ Proved
- \[ = - \sin{\left(x + 2 \right)} \]trigThe derivative of cos(u) is -sin(u).✓ Proved
- \[ = - \sin{\left(x \right)} \cos{\left(2 \right)} - \sin{\left(2 \right)} \cos{\left(x \right)} \]algebraExpand using the sine addition formula.✓ Proved
Answer \( - \sin{\left(x + 2 \right)} \)
✓ 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 |
| 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 1 applies the derivative of cos(u) but omits the chain rule for the inner function u=x+2. The correct first derivative should be -sin(x+2)·1, i.e. -sin(x+2).qwen3.6:27b-mlx: fail (error) — Step 2 is mathematically incorrect because it omits the chain rule factor (the derivative of the inner function x+2), resulting in an incomplete derivative. Step 3 incorrectly applies the sine addition formula to -sin(x+2) instead of sin(x+2), introducing a sign error.
Senior review claude-sonnet-5-5, 2026-10-08: pass — All lines are correct and SymPy-verified. The derivative of cos(x+2) is -sin(x+2), where the inner derivative 1 is absorbed in the trig/chain step, and the expansion in line 3 has correct signs.
gpt-oss:20b: dismiss — The derivative of x+2 is 1, so -sin(x+2) is exactly the chain-rule result; the objection even concedes the value is -sin(x+2), and the 'trig' label is legitimate.qwen3.6:27b-mlx: dismiss — The chain-rule factor is 1, so line 2 is correct (SymPy proved it), and line 3 correctly expands -sin(x+2) as -(sin x cos 2 + cos x sin 2), so there is no sign error (also SymPy-proved).
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-08 — All lines are correct and SymPy-verified. The derivative of cos(x+2) is -sin(x+2), where the inner derivative 1 is absorbed in the trig/chain step, and the expansion in line 3 has correct signs.qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 2 is mathematically incorrect because it omits the chain rule factor (the derivative of the inner function x+2), resulting in an incomplete derivative. Step 3 incorrectly applies the sine addition formula to -sin(x+2) instead of sin(x+2), introducing a sign error.gpt-oss:20b: fail (error) 2026-09-21 — Step 1 applies the derivative of cos(u) but omits the chain rule for the inner function u=x+2. The correct first derivative should be -sin(x+2)·1, i.e. -sin(x+2).qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 2 is mathematically incorrect because it omits the chain rule factor (the derivative of the inner function x+2). Step 3 is also incorrect because it applies the sine addition formula to -sin(x+2) instead of the correct derivative -sin(x+2).gpt-oss:20b: fail (style) 2026-09-20 — Step 1 lacks a proper rule label (should be "chain"), and step 3 is labeled "algebra" but actually applies the sine addition formula, which belongs to the "trig" rule. These labeling errors affect the granularity of the solution.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 2 claims the derivative of cos(x+2) is -sin(x+2) using the 'trig' rule, but this omits the chain rule factor (the derivative of the inner function x+2). The step applies both the trigonometric derivative and the chain rule simultaneously, violating the one-rule-per-step constraint, and the label 'trig' does not account for the missing chain rule application.gpt-oss:20b: fail (error) 2026-09-20 — Step 2 applies the derivative of cos(u) but omits the chain‑rule factor d(u)/dx. Step 3 uses the sine addition formula, which is a trig identity, not an algebraic simplification, yet it is labeled "algebra". Both steps violate the rule‑granularity 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 2026-09-26 with SymPy 1.14.0.