Derivative of \( \displaystyle \frac{\cos{\left(x + 2 \right)}}{x + 2} \)
Problem 2.504 · hard
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(x + 2 \right)}}{x + 2} \).
- \[ \frac{d}{d x} \frac{\cos{\left(x + 2 \right)}}{x + 2} \]rewriteStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
- \[ = \cos{\left(x + 2 \right)} \frac{d}{d x} \frac{1}{x + 2} + \frac{\frac{d}{d x} \cos{\left(x + 2 \right)}}{x + 2} \]productApply the product rule.✓ Proved
- \[ = \cos{\left(x + 2 \right)} \frac{d}{d x} \frac{1}{x + 2} - \frac{\sin{\left(x + 2 \right)}}{x + 2} \]trigDifferentiate the first part of the product.✓ Proved
- \[ = - \frac{\sin{\left(x + 2 \right)}}{x + 2} - \frac{\cos{\left(x + 2 \right)}}{\left(x + 2\right)^{2}} \]chain derivative algebraDifferentiate the second part using the chain rule. Differentiate the power term. Simplify the expression by distributing the terms.✓ Proved
- \[ = \frac{- \left(x + 2\right) \sin{\left(x + 2 \right)} - \cos{\left(x + 2 \right)}}{\left(x + 2\right)^{2}} \]algebraCombine the terms over a common denominator.✓ Proved
Answer \( - \frac{\sin{\left(x + 2 \right)}}{x + 2} - \frac{\cos{\left(x + 2 \right)}}{\left(x + 2\right)^{2}} \)
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 undefined where x + 2 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 2 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where x + 2 = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: fail (style) — Step 6 applies a label "derivative" to a simplification step that merely removes the extraneous factor 1 and rewrites the expression; the correct label should be "simplify".qwen3.6:27b-mlx: fail (error) — Step 4 applies the chain rule to differentiate cos(x+2) but is labeled 'trig'. Step 5 applies the chain rule to differentiate (x+2)^(-1) but is labeled 'chain', while Step 6 is labeled 'derivative' for a simplification step that does not perform differentiation. The labeling is inconsistent and incorrect regarding the rules applied.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 4 applies the chain rule to differentiate cos(x+2) but is labeled 'trig'. Step 5 applies the chain rule to differentiate (x+2)^(-1) but is labeled 'chain', while Step 6 is labeled 'derivative' for a simplification step that does not perform differentiation. The labeling is inconsistent and incorrect regarding the rules applied.gpt-oss:20b: fail (style) 2026-09-21 — Step 6 applies a label "derivative" to a simplification step that merely removes the extraneous factor 1 and rewrites the expression; the correct label should be "simplify".qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 5 is labeled 'chain' but performs the differentiation of the power term (which should be labeled 'power') and the inner derivative (which should be labeled 'derivative') simultaneously. Step 6 is labeled 'derivative' but performs only algebraic simplification (multiplying by 1), which is a mislabeling of the operation performed.gpt-oss:20b: fail (error) 2026-09-20 — Step 4 applies the chain rule to differentiate cos(x+2) but labels it only as "trig". The derivative of cos(u) is -sin(u)*u', so the chain rule must be invoked. The label is incorrect and the step is not fully justified by the named rule.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 5 applies the chain rule to differentiate (x+2)^(-1), but Step 6 is redundant and mislabeled as 'derivative' when it merely simplifies the result of Step 5 (multiplying by 1). More critically, Step 5's note claims to differentiate the second part using the chain rule, but the step itself shows the full differentiation result including the inner derivative '1'. Step 6 then removes this '1'. This splits a single logical differentiation step into two, violating the 'one rule per step' principle if interpreted strictly, or is just redundant. However, the primary defect is that Step 6 is labeled 'derivative' but performs no differentiation; it is an algebraic simplification (removing *1). The label 'derivative' is incorrect for a step that does not apply a differentiation rule.gpt-oss:20b: fail (style) 2026-09-20 — Step 6 is labeled "derivative" but it only simplifies the expression; the correct label would be "simplify" or "algebra". This mislabeling could mislead a student about the rule applied.
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.