Derivative of \( \displaystyle e^{3 x - 3} \)
Problem 2.382 · easy Video
- \[ \frac{d}{d x} e^{3 x - 3} \]Start with the derivative of the function.✓ Proved
- \[ = e^{3 x - 3} \frac{d}{d x} \left(3 x - 3\right) \]chainApply the chain rule.✓ Proved
- \[ = - e^{3 x - 3} \frac{d}{d x} 3 + e^{3 x - 3} \frac{d}{d x} 3 x \]sumDifferentiate the terms inside the parenthesis.✓ Proved
- \[ = 3 e^{3 x - 3} \]derivative simplifyDifferentiate the linear and constant terms. Simplify the expression.✓ Proved
✓ 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 |
| 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 (style) — Step 3 applies two rules at once (splitting the inner derivative and distributing the exponential factor) and incorrectly labels the operation as "sum".qwen3.6:27b-mlx: pass — The solution correctly applies the chain rule, linearity of the derivative, and basic differentiation rules in separate steps. The labels are appropriate for the operations performed.
Senior review claude-sonnet-5-5, 2026-10-08: pass — Every line is SymPy-verified and the rule labels are legitimate. The chain rule gives exp(3x-3)*3, and the constant term's derivative is 0, so the answer 3*exp(3*x-3) is correct.
gpt-oss:20b: dismiss — Step 3 applies the sum rule to the inner derivative, and distributing the exponential factor across the resulting terms is part of that same change, so 'sum' is a legitimate label and only one thing changes.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-08 — Every line is SymPy-verified and the rule labels are legitimate. The chain rule gives exp(3x-3)*3, and the constant term's derivative is 0, so the answer 3*exp(3*x-3) is correct.qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the chain rule, linearity of the derivative, and basic differentiation rules in separate steps. The labels are appropriate for the operations performed.gpt-oss:20b: fail (style) 2026-09-21 — Step 3 applies two rules at once (splitting the inner derivative and distributing the exponential factor) and incorrectly labels the operation as "sum".qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 is mathematically incorrect because it distributes the outer exponential term into the derivative of the sum, effectively applying the product rule to a constant multiplier which is not present in the previous step. The chain rule result in Step 2 is a single product term, so the derivative of the inner function should be computed as a whole, not split by distributing the outer factor.gpt-oss:20b: fail (error) 2026-09-20 — Step 3 applies two rules at once: it distributes the factor exp(3*x‑3) over the sum 3*x‑3, which is an algebraic distributive step, and then uses the sum rule for derivatives. Each step must change only one thing, so this step is invalid.
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.