Derivative of \( \displaystyle \left(x + 2\right)^{2} e^{- x - 2} \)
Problem 2.451 · hard Video
Differentiate \( \displaystyle f(x) = \left(x + 2\right)^{2} e^{- x - 2} \).
- \[ \frac{d}{d x} \left(x + 2\right)^{2} e^{- x - 2} \]Start with the derivative of the function.✓ Proved
- \[ = \left(x + 2\right)^{2} \frac{d}{d x} e^{- x - 2} + e^{- x - 2} \frac{d}{d x} \left(x + 2\right)^{2} \]productApply the product rule.✓ Proved
- \[ = \left(x + 2\right)^{2} e^{- x - 2} \frac{d}{d x} \left(- x - 2\right) + e^{- x - 2} \frac{d}{d x} \left(x + 2\right)^{2} \]chainApply the chain rule to the exponential term.✓ Proved
- \[ = - \left(x + 2\right)^{2} e^{- x - 2} + \left(2 x + 4\right) e^{- x - 2} \]derivative algebraDifferentiate the inner function and the power term. Simplify the expression.✓ Proved
- \[ = - x \left(x + 2\right) e^{- x - 2} \]algebra algebra algebraFactor out the common terms. Simplify the term in the parentheses. Rearrange the terms for the final answer.✓ Proved
Answer \( x \left(- x - 2\right) e^{- x - 2} \)
✓ 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 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ 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 (error) — Step 4 applies two differentiation rules simultaneously (derivative of the inner function and derivative of the power term), violating the one-change-per-step constraint. Additionally, the label 'derivative' is incorrect for the simplification of the first term's coefficient (-1), which is algebraic.
Senior review claude-sonnet-5-5, 2026-10-08: pass — The product rule, chain rule, and algebra steps are all correct, and SymPy proved every line and the final answer. The labels are legitimate, and step 4 only evaluates the pending derivatives, so no defect remains.
qwen3.6:27b-mlx: dismiss — Step 4 evaluates the remaining Derivative objects (the inner linear function and the power term), which the contract allows as one 'derivative' step, and the label names a rule that was actually applied; SymPy proved the line.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-08 — The product rule, chain rule, and algebra steps are all correct, and SymPy proved every line and the final answer. The labels are legitimate, and step 4 only evaluates the pending derivatives, so no defect remains.qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 4 applies two differentiation rules simultaneously (derivative of the inner function and derivative of the power term), violating the one-change-per-step constraint. Additionally, the label 'derivative' is incorrect for the simplification of the first term's coefficient (-1), which is algebraic.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 4 applies two distinct differentiation rules simultaneously (differentiating the inner function of the exponential and the power term) under a single 'derivative' label, violating the one-rule-per-step constraint.gpt-oss:20b: fail (error) 2026-09-20 — Step 4 applies two derivative rules at once—d/dx(-x-2) and d/dx((x+2)**2)—in a single line, violating the one‑change‑per‑step rule.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 4 applies the derivative rule to two distinct terms simultaneously (the inner function of the exponential and the power term), violating the one-change-per-step constraint.gpt-oss:20b: pass 2026-09-20
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.