Derivative of \( \displaystyle \left(5 x - 2\right) e^{5 x - 1} \)
Problem 2.982 · medium Video
Differentiate \( \displaystyle f(x) = \left(5 x - 2\right) e^{5 x - 1} \).
- \[ \frac{d}{d x} \left(5 x - 2\right) e^{5 x - 1} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(5 x e^{5 x - 1} - 2 e^{5 x - 1}\right) \]algebraDistribute the term.✓ Proved
- \[ = \frac{d}{d x} 5 x e^{5 x - 1} - \frac{d}{d x} 2 e^{5 x - 1} \]sumApply the sum rule.✓ Proved
- \[ = 5 \frac{d}{d x} x e^{5 x - 1} - 2 \frac{d}{d x} e^{5 x - 1} \]constant-multipleFactor out the constants.✓ Proved
- \[ = 5 x \frac{d}{d x} e^{5 x - 1} + 5 e^{5 x - 1} \frac{d}{d x} x - 2 \frac{d}{d x} e^{5 x - 1} \]productApply the product rule to the first term.✓ Proved
- \[ = 5 x \frac{d}{d x} e^{5 x - 1} + 5 e^{5 x - 1} - 2 \frac{d}{d x} e^{5 x - 1} \]derivativeDifferentiate x.✓ Proved
- \[ = 25 x e^{5 x - 1} - 5 e^{5 x - 1} \]chain product algebraDifferentiate the exponential term using the chain rule. Distribute the 5 into the parentheses. Combine like terms.✓ Proved
- \[ = \left(25 x - 5\right) e^{5 x - 1} \]simplifyFactor out the common term.✓ Proved
Answer \( \left(25 x - 5\right) e^{5 x - 1} \)
✓ 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 |
| 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: fail (style) — Step 8 is labeled 'product' but performs algebraic distribution (multiplying a constant into a sum), which should be labeled 'algebra'. Step 7 applies the chain rule to both instances of exp(5x-1) simultaneously, violating the one-rule-per-step constraint.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-09-28 — Step 8 is labeled 'product' but performs algebraic distribution (multiplying a constant into a sum), which should be labeled 'algebra'. Step 7 applies the chain rule to both instances of exp(5x-1) simultaneously, violating the one-rule-per-step constraint.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (style) 2026-09-27 — Step 8 is labeled 'product' but performs algebraic distribution (multiplying a constant into a sum), which should be labeled 'algebra'. Step 7 is labeled 'chain' but effectively applies the exponential derivative rule (which implies the chain rule); while 'chain' is acceptable for the inner derivative, the step also simplifies the coefficient, making it slightly ambiguous, but the primary defect is the mislabeling in Step 8.gpt-oss:20b: pass 2026-09-27 — Step 8 is labeled "product" but it is a simple algebraic distribution of the constant 5 into the parentheses; the correct label would be "algebra".qwen3.6:27b-mlx: fail (style) 2026-09-27 — Step 8 is labeled 'product' but performs algebraic distribution (multiplying a constant into a sum), which should be labeled 'algebra'. Step 10 is labeled 'simplify' but performs factoring, which is better described as 'algebra' or 'rewrite'.gpt-oss:20b: pass 2026-09-27
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 2026-09-28 with SymPy 1.14.0.