Derivative of \( \displaystyle \left(3 x - 3\right) e^{2 x - 1} \)
Problem 2.1141 · medium Video
Differentiate \( \displaystyle f(x) = 3 \left(x - 1\right) e^{2 x - 1} \).
- \[ \frac{d}{d x} \left(3 x - 3\right) e^{2 x - 1} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 3 \frac{d}{d x} \left(x - 1\right) e^{2 x - 1} \]constant-multiplePull out the constant factor 3.✓ Proved
- \[ = 3 \left(x - 1\right) \frac{d}{d x} e^{2 x - 1} + 3 e^{2 x - 1} \frac{d}{d x} \left(x - 1\right) \]productApply the product rule.✓ Proved
- \[ = 3 \left(x - 1\right) \frac{d}{d x} e^{2 x - 1} + 3 \left(- \frac{d}{d x} 1 + \frac{d}{d x} x\right) e^{2 x - 1} \]sumDifferentiate the terms inside the parenthesis separately.✓ Proved
- \[ = 3 \left(x - 1\right) \frac{d}{d x} e^{2 x - 1} + 3 e^{2 x - 1} \]derivative simplifyEvaluate the derivatives of x and 1. Simplify the expression inside the parentheses.✓ Proved
- \[ = 3 \left(x - 1\right) e^{2 x - 1} \frac{d}{d x} \left(2 x - 1\right) + 3 e^{2 x - 1} \]chainApply the chain rule to the exponential term.✓ Proved
- \[ = 6 \left(x - 1\right) e^{2 x - 1} + 3 e^{2 x - 1} \]derivativeEvaluate the derivative of the exponent.✓ Proved
- \[ = 3 \left(2 x - 2\right) e^{2 x - 1} + 3 e^{2 x - 1} \]algebraRearrange the terms.✓ Proved
- \[ = 3 \left(2 x - 1\right) e^{2 x - 1} \]algebra algebra simplifyFactor out the common exponential term. Distribute the 2 into the parenthesis. Combine like terms to get the final answer.✓ Proved
Answer \( \left(6 x - 3\right) e^{2 x - 1} \)
✓ 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. 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 |
| 12 | ✓ 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 constant multiple, product, and chain rules in separate steps. All labels correspond to 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 constant multiple, product, and chain rules in separate steps. All labels correspond to the operations performed, and the algebraic simplifications are valid.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-28gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-28gpt-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.