Derivative of \( \displaystyle \left(2 x - 1\right)^{2} e^{1 - 2 x} \)
Problem 2.612 · hard Video
Differentiate \( \displaystyle f(x) = \left(2 x - 1\right)^{2} e^{1 - 2 x} \).
- \[ \frac{d}{d x} \left(2 x - 1\right)^{2} e^{1 - 2 x} \]Start with the derivative of the function.✓ Proved
- \[ = \left(2 x - 1\right)^{2} \frac{d}{d x} e^{1 - 2 x} + e^{1 - 2 x} \frac{d}{d x} \left(2 x - 1\right)^{2} \]productApply the product rule.✓ Proved
- \[ = \left(2 x - 1\right)^{2} \frac{d}{d x} e^{1 - 2 x} + \left(4 x - 2\right) e^{1 - 2 x} \frac{d}{d x} \left(2 x - 1\right) \]powerDifferentiate the first part using the power rule.✓ Proved
- \[ = \left(2 x - 1\right)^{2} \frac{d}{d x} e^{1 - 2 x} + \left(8 x - 4\right) e^{1 - 2 x} \]derivative algebraDifferentiate the inner part of the power term. Simplify the first term.✓ Proved
- \[ = \left(2 x - 1\right)^{2} e^{1 - 2 x} \frac{d}{d x} \left(1 - 2 x\right) + \left(8 x - 4\right) e^{1 - 2 x} \]chainApply the chain rule to the exponential term.✓ Proved
- \[ = - 2 \left(2 x - 1\right)^{2} e^{1 - 2 x} + \left(8 x - 4\right) e^{1 - 2 x} \]derivative algebraDifferentiate the exponent. Simplify the expression.✓ Proved
- \[ = \left(6 - 4 x\right) \left(2 x - 1\right) e^{1 - 2 x} \]algebra algebra simplifyFactor out common terms. Distribute the -2 inside the parentheses. Combine like terms.✓ Proved
Answer \( \left(- 8 x^{2} + 16 x - 6\right) e^{1 - 2 x} \)
✓ 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 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 11 | ✓ 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, power, chain, and derivative rules in separate steps. The algebraic simplifications are correct and lead to the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the product, power, chain, and derivative rules in separate steps. The algebraic simplifications are correct and lead to the stated answer.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the product, power, chain, and derivative rules in a logical sequence. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the product, power, chain, and derivative rules in a logical sequence. Each step isolates a single transformation, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-21
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.