Derivative of \( \displaystyle \left(x + 1\right) e^{x + 2} \)
Problem 2.403 · medium
Differentiate \( \displaystyle f(x) = \left(x + 1\right) e^{x + 2} \).
- \[ \frac{d}{d x} \left(x + 1\right) e^{x + 2} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \left(x + 1\right) \frac{d}{d x} e^{x + 2} + e^{x + 2} \frac{d}{d x} \left(x + 1\right) \]productApply the product rule.✓ Proved
- \[ = \left(x + 1\right) \frac{d}{d x} e^{x + 2} + e^{x + 2} \]derivative constantDifferentiate the first part of the product. The derivative of x + 1 is 1.✓ Proved
- \[ = \left(x + 1\right) e^{x + 2} \frac{d}{d x} \left(x + 2\right) + e^{x + 2} \]chainApply the chain rule to the exponential term.✓ Proved
- \[ = \left(x + 1\right) e^{x + 2} + e^{x + 2} \]derivative constantDifferentiate the exponent x + 2. Simplify the derivative of the exponent.✓ Proved
- \[ = \left(x + 2\right) e^{x + 2} \]algebra simplifyFactor out the common term exp(x + 2). Simplify the expression inside the parentheses.✓ Proved
Answer \( \left(x + 2\right) e^{x + 2} \)
✓ 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.
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 |
| 9 | ✓ 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 (style) — Step 4 incorrectly labels the derivative of x+1 as "constant"; it should be "derivative". Step 7 also mislabels a simplification as "constant".qwen3.6:27b-mlx: pass — The solution correctly applies the product rule, chain rule, and basic differentiation rules in a step-by-step manner. Each step changes only one aspect of the expression and uses an appropriate label from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the product rule, chain rule, and basic differentiation rules in a step-by-step manner. Each step changes only one aspect of the expression and uses an appropriate label from the fixed vocabulary.gpt-oss:20b: fail (style) 2026-09-21 — Step 4 incorrectly labels the derivative of x+1 as "constant"; it should be "derivative". Step 7 also mislabels a simplification as "constant".qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the product rule, chain rule, and basic differentiation rules in a step-by-step manner. Each step changes only one aspect of the expression and uses an appropriate label from the fixed vocabulary.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the product rule, chain rule, and basic differentiation rules in separate steps. The labels used are consistent with the provided vocabulary and accurately describe the operations performed.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.