Derivative of \( \displaystyle \left(- x - \frac{1}{5}\right) e^{5 x + 2} \)
Problem 2.668 · medium
Differentiate \( \displaystyle f(x) = \left(- x - \frac{1}{5}\right) e^{5 x + 2} \).
- \[ \frac{d}{d x} \left(- x - \frac{1}{5}\right) e^{5 x + 2} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \left(- x - \frac{1}{5}\right) \frac{d}{d x} e^{5 x + 2} + e^{5 x + 2} \frac{d}{d x} \left(- x - \frac{1}{5}\right) \]productApply the product rule.✓ Proved
- \[ = \left(- x - \frac{1}{5}\right) \frac{d}{d x} e^{5 x + 2} + e^{5 x + 2} \frac{d}{d x} \left(- \frac{1}{5}\right) + e^{5 x + 2} \frac{d}{d x} \left(- x\right) \]sumSplit the derivative of the first term.✓ Proved
- \[ = \left(- x - \frac{1}{5}\right) \frac{d}{d x} e^{5 x + 2} - e^{5 x + 2} \]derivative algebraDifferentiate the individual terms. Simplify the expression by removing the zero term.✓ Proved
- \[ = \left(- x - \frac{1}{5}\right) e^{5 x + 2} \frac{d}{d x} \left(5 x + 2\right) - e^{5 x + 2} \]chainApply the chain rule to the exponential term.✓ Proved
- \[ = 5 \left(- x - \frac{1}{5}\right) e^{5 x + 2} - e^{5 x + 2} \]derivativeDifferentiate the inner function of the exponent.✓ Proved
- \[ = \left(- 5 x - 1\right) e^{5 x + 2} - e^{5 x + 2} \]algebraDistribute the constant 5.✓ Proved
- \[ = \left(- 5 x - 2\right) e^{5 x + 2} \]algebra simplify simplifyFactor out the common exponential term. Combine like terms. Final simplified form.✓ Proved
Answer \( \left(- 5 x - 2\right) e^{5 x + 2} \)
✓ 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.
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 rule, chain rule, and algebraic simplifications in a step-by-step manner. Each step changes only one aspect of the expression and uses appropriate labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the product rule, chain rule, and algebraic simplifications in a step-by-step manner. Each step changes only one aspect of the expression and uses appropriate labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the product rule, chain rule, and algebraic simplifications. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.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.