Derivative of \( \displaystyle \left(x - 2\right) e^{x - 1} \)
Problem 2.1361 · medium
Differentiate \( \displaystyle f(x) = \left(x - 2\right) e^{x - 1} \).
- \[ \frac{d}{d x} \left(x - 2\right) e^{x - 1} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \left(x - 2\right) \frac{d}{d x} e^{x - 1} + e^{x - 1} \frac{d}{d x} \left(x - 2\right) \]productApply the product rule.✓ Proved
- \[ = \left(x - 2\right) \frac{d}{d x} e^{x - 1} + e^{x - 1} \]derivative constantDifferentiate the first part (x - 2). The derivative of x is 1.✓ Proved
- \[ = \left(x - 2\right) e^{x - 1} \frac{d}{d x} \left(x - 1\right) + e^{x - 1} \]chainApply the chain rule to exp(x - 1).✓ Proved
- \[ = \left(x - 2\right) e^{x - 1} + e^{x - 1} \]derivative constantDifferentiate the inner function (x - 1). The derivative of x - 1 is 1.✓ Proved
- \[ = \left(x - 1\right) e^{x - 1} \]algebra simplifyFactor out the common term exp(x - 1). Simplify the expression inside the parentheses.✓ Proved
Answer \( \left(x - 1\right) e^{x - 1} \)
✓ 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 applies a simplification (1*exp(x-1) → exp(x-1)) but labels it as "constant", which is not an appropriate rule name from the allowed vocabulary.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: fail (style) 2026-10-03 — Step 4 applies a simplification (1*exp(x-1) → exp(x-1)) but labels it as "constant", which is not an appropriate rule name from the allowed vocabulary.qwen3.6:27b-mlx: pass 2026-09-30gpt-oss:20b: fail (error) 2026-09-30 — Step 1 incorrectly labels the application of the product rule as "derivative"; it should be "product". Step 4 uses the label "constant" for simplifying 1·exp(x‑1) to exp(x‑1), which is an algebraic simplification, not a constant rule. Step 7 similarly mislabels the simplification of 1 as a "constant" rule; it should be "simplify" or "algebra".
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.