Derivative of \( \displaystyle \left(3 x - \frac{3}{2}\right) e^{4 x - 1} \)
Problem 2.1381 · medium
Differentiate \( \displaystyle f(x) = \left(3 x - \frac{3}{2}\right) e^{4 x - 1} \).
- \[ \frac{d}{d x} \left(3 x - \frac{3}{2}\right) e^{4 x - 1} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(3 x e^{4 x - 1} - \frac{3 e^{4 x - 1}}{2}\right) \]algebraDistribute the terms.✓ Proved
- \[ = \frac{d}{d x} 3 x e^{4 x - 1} - \frac{d}{d x} \frac{3 e^{4 x - 1}}{2} \]sumApply the sum rule for differentiation.✓ Proved
- \[ = 3 \frac{d}{d x} x e^{4 x - 1} - \frac{3 \frac{d}{d x} e^{4 x - 1}}{2} \]constant-multiplePull out the constant factors.✓ Proved
- \[ = 3 x \frac{d}{d x} e^{4 x - 1} + 3 e^{4 x - 1} \frac{d}{d x} x - \frac{3 \frac{d}{d x} e^{4 x - 1}}{2} \]productApply the product rule to the first term.✓ Proved
- \[ = 3 x \frac{d}{d x} e^{4 x - 1} + 3 e^{4 x - 1} - \frac{3 \frac{d}{d x} e^{4 x - 1}}{2} \]derivativeDifferentiate x.✓ Proved
- \[ = 3 x e^{4 x - 1} \frac{d}{d x} \left(4 x - 1\right) + 3 e^{4 x - 1} - \frac{3 \frac{d}{d x} e^{4 x - 1}}{2} \]Wait, let's use the chain rule directly.✓ Proved
- \[ = 12 x e^{4 x - 1} - 3 e^{4 x - 1} \]chain product algebraApply the chain rule to the exponential term. Distribute the 3 and evaluate the derivative of the second term. Combine like terms.✓ Proved
Answer \( \left(12 x - 3\right) e^{4 x - 1} \)
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. A reviewer's objection to the explanation stands; see the receipt.
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 |
| 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 7 applies two transformations at once (an unnecessary algebraic rewrite plus the chain rule), violating the one‑rule‑per‑step rule. Step 9 is labeled "product" but is merely an algebraic distribution and simplification, so the label is incorrect.qwen3.6:27b-mlx: fail (error) — Step 7 is mathematically nonsensical, introducing a division by (4*x - 1) and multiplying by it in a way that does not follow from the previous line or standard differentiation rules. Step 9 applies two operations (distributing the constant 3 and evaluating the derivative of the second term) in a single step, violating the one-rule-per-step constraint.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 7 is mathematically nonsensical, introducing a division by (4*x - 1) and multiplying by it in a way that does not follow from the previous line or standard differentiation rules. Step 9 applies two operations (distributing the constant 3 and evaluating the derivative of the second term) in a single step, violating the one-rule-per-step constraint.gpt-oss:20b: fail (error) 2026-10-03 — Step 7 applies two transformations at once (an unnecessary algebraic rewrite plus the chain rule), violating the one‑rule‑per‑step rule. Step 9 is labeled "product" but is merely an algebraic distribution and simplification, so the label is incorrect.qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 7 contains nonsensical algebraic manipulation (multiplying by the argument and its reciprocal) and is labeled 'unlabelled', which is not in the allowed vocabulary. Step 9 applies both the product rule (distribution) and the chain rule (evaluating the derivative of the second term) simultaneously, violating the one-rule-per-step constraint.gpt-oss:20b: fail (error) 2026-10-03 — Step 7 applies two rules at once (an incorrect chain‑rule manipulation and a simplification) and is labeled unlabelled. This violates the one‑rule‑per‑step rule and contains an algebraic error in the derivative of exp(4*x‑1).
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.