Derivative of \( \displaystyle \left(\frac{1}{3} - x\right) e^{3 x} \)
Problem 2.551 · medium
Differentiate \( \displaystyle f(x) = \left(\frac{1}{3} - x\right) e^{3 x} \).
- \[ \frac{d}{d x} \left(\frac{1}{3} - x\right) e^{3 x} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \left(\frac{1}{3} - x\right) \frac{d}{d x} e^{3 x} + e^{3 x} \frac{d}{d x} \left(\frac{1}{3} - x\right) \]productApply the product rule.✓ Proved
- \[ = 3 \left(\frac{1}{3} - x\right) e^{3 x} - e^{3 x} \]derivativeDifferentiate each part of the product.✓ Proved
- \[ = \left(1 - 3 x\right) e^{3 x} - e^{3 x} \]algebra algebraSimplify the coefficients. Distribute the 3 into the parentheses.✓ Proved
- \[ = - 3 x e^{3 x} \]algebra simplifyFactor out the common exponential term. Simplify the remaining expression.✓ Proved
Answer \( - 3 x e^{3 x} \)
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. The reviewers disagree about how one step is explained; every verdict is in 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 |
| 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 (implicitly within the derivative steps), and algebraic simplification. Each step changes only one aspect of the expression and uses valid 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 (implicitly within the derivative steps), and algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 applies both the chain rule (for exp(3x)) and the power/constant rules (for 1/3 - x) simultaneously, violating the one-rule-per-step constraint. It should be split into separate steps for each differentiation.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.