Derivative of \( \displaystyle \left(x - 3\right) e^{3 - x} \)
Problem 2.436 · hard Video
Differentiate \( \displaystyle f(x) = \left(x - 3\right) e^{3 - x} \).
- \[ \frac{d}{d x} \left(x - 3\right) e^{3 - x} \]Start with the derivative of the function.✓ Proved
- \[ = \left(x - 3\right) \frac{d}{d x} e^{3 - x} + e^{3 - x} \frac{d}{d x} \left(x - 3\right) \]productApply the product rule.✓ Proved
- \[ = \left(x - 3\right) e^{3 - x} \frac{d}{d x} \left(3 - x\right) + e^{3 - x} \]derivativeDifferentiate the first part using the chain rule.✓ Proved
- \[ = - \left(x - 3\right) e^{3 - x} + e^{3 - x} \]derivativeDifferentiate the remaining terms.✓ Proved
- \[ = \left(3 - x\right) e^{3 - x} + e^{3 - x} \]algebraDistribute the negative sign.✓ Proved
- \[ = \left(4 - x\right) e^{3 - x} \]algebra algebra simplifyFactor out the common exponential term. Simplify the expression inside the parentheses. Final simplified form.✓ Proved
Answer \( \left(4 - x\right) e^{3 - x} \)
✓ 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: fail (error) — Step 3 applies two derivative rules at once (chain rule for exp(3‑x) and derivative of x‑3). Each step must change only one thing, so this is a defect.qwen3.6:27b-mlx: fail (error) — Step 3 applies the chain rule to the exponential term but labels the step 'derivative', which is incorrect; the label should be 'chain'. Furthermore, the note claims to differentiate the 'first part' using the chain rule, but the step actually differentiates both terms of the product sum simultaneously, violating the one-rule-per-step constraint.
Senior review claude-sonnet-5-5, 2026-10-08: fail — The math is correct throughout, and SymPy proved every line and the final answer. Step 3 combines the chain-rule expansion of the first term with evaluating d/dx(x-3) in the second, and its note mentions only the first part. This is a granularity and phrasing issue, not an error.
gpt-oss:20b: uphold — Step 3 applies the chain rule to the exp(3-x) term and also evaluates Derivative(x-3) as 1-0 in the other term; these are separate differentiation rules, so it is two changes in one step.qwen3.6:27b-mlx: dismiss — 'derivative' is in the vocabulary and is a legitimate label for evaluating derivatives, so the label is not wrong, and 'chain' would be an equally acceptable alternative. The objection's second point, that two terms are changed at once, duplicates the gpt-oss one and is upheld there.
Every verdict on record (7)
claude-sonnet-5-5: fail (style) 2026-10-08 — The math is correct throughout, and SymPy proved every line and the final answer. Step 3 combines the chain-rule expansion of the first term with evaluating d/dx(x-3) in the second, and its note mentions only the first part. This is a granularity and phrasing issue, not an error.qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 applies the chain rule to the exponential term but labels the step 'derivative', which is incorrect; the label should be 'chain'. Furthermore, the note claims to differentiate the 'first part' using the chain rule, but the step actually differentiates both terms of the product sum simultaneously, violating the one-rule-per-step constraint.gpt-oss:20b: fail (error) 2026-09-21 — Step 3 applies two derivative rules at once (chain rule for exp(3‑x) and derivative of x‑3). Each step must change only one thing, so this is a defect.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 applies both the chain rule (for exp(3-x)) and the power/constant rule (for x-3) simultaneously, violating the one-rule-per-step constraint. Additionally, the label 'derivative' is incorrect for applying the chain rule, which is a distinct rule in the vocabulary.gpt-oss:20b: fail (error) 2026-09-20 — Step 3 applies two derivative rules at once (chain rule for exp(3‑x) and derivative of x‑3). Each step must change only one thing, so this is a defect.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 is labeled 'derivative' but applies the chain rule to the exponential term; the label 'chain' is required. Additionally, Step 3 performs two distinct operations (differentiating the exponential via chain rule and differentiating the polynomial term) in a single step, violating the one-rule-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.