Derivative of \( \displaystyle \cos{\left(x + 1 \right)} \)
Problem 2.1202 · easy Mental math
Differentiate \( \displaystyle f(x) = \cos{\left(x + 1 \right)} \).
- \[ \frac{d}{d x} \cos{\left(x + 1 \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \sin{\left(x + 1 \right)} \frac{d}{d x} \left(x + 1\right) \]chainApply the chain rule.✓ Proved
- \[ = - \sin{\left(x + 1 \right)} \]derivative simplifyDifferentiate the inner function. Simplify the expression.✓ Proved
Answer \( - \sin{\left(x + 1 \right)} \)
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 |
| 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 (misleading) — Step 1 incorrectly labels the application of the chain rule as "derivative"; step 2 incorrectly labels the differentiation of the inner function as "chain". These mislabelings could teach a student that the chain rule is a separate rule from the derivative of cosine, which is misleading.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29gpt-oss:20b: fail (misleading) 2026-09-29 — Step 1 incorrectly labels the application of the chain rule as "derivative"; step 2 incorrectly labels the differentiation of the inner function as "chain". These mislabelings could teach a student that the chain rule is a separate rule from the derivative of cosine, which is misleading.qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the chain rule and simplifies the result. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-29
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 2026-09-29 with SymPy 1.14.0.