Derivative of \( \displaystyle \left(2 x + 2\right) e^{x + 2} \)
Problem 2.1127 · medium Video
Differentiate \( \displaystyle f(x) = 2 \left(x + 1\right) e^{x + 2} \).
- \[ \frac{d}{d x} \left(2 x + 2\right) e^{x + 2} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \left(2 x + 2\right) \frac{d}{d x} e^{x + 2} + 2 e^{x + 2} \frac{d}{d x} \left(x + 1\right) \]productApply the product rule.✓ Proved
- \[ = \left(2 x + 2\right) \frac{d}{d x} e^{x + 2} + 2 e^{x + 2} \]derivative constantDifferentiate the first part of the product. Simplify the derivative of (x + 1).✓ Proved
- \[ = \left(2 x + 2\right) e^{x + 2} \frac{d}{d x} \left(x + 2\right) + 2 e^{x + 2} \]chainApply the chain rule to the exponential term.✓ Proved
- \[ = \left(2 x + 2\right) e^{x + 2} + 2 e^{x + 2} \]derivative algebraDifferentiate the exponent (x + 2). Simplify the expression.✓ Proved
- \[ = 2 \left(x + 2\right) e^{x + 2} \]algebra simplifyFactor out the common term 2*exp(x + 2). Simplify the final expression.✓ Proved
Answer \( 2 \left(x + 2\right) e^{x + 2} \)
✓ 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 |
| 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: passqwen3.6:27b-mlx: pass — The solution correctly applies the product rule, chain rule, and basic differentiation rules in separate steps. The labels match the operations performed, and the algebraic simplifications are valid.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the product rule, chain rule, and basic differentiation rules in separate steps. The labels match the operations performed, and the algebraic simplifications are valid.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the product rule, chain rule, and basic differentiation rules in a logical sequence. Each step changes only one aspect of the expression and uses appropriate labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the product rule, chain rule, and algebraic simplification in distinct steps. Each label accurately reflects the operation performed.gpt-oss:20b: pass 2026-09-28
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 2026-10-03 with SymPy 1.14.0.