Derivative of \( \displaystyle e^{3 x - 3} \)
Problem 2.382 · easy
Differentiate \( \displaystyle f(x) = e^{3 x - 3} \).
- \[ \frac{d}{d x} e^{3 x - 3} \]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
Answer \( 3 e^{3 x - 3} \)
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 |
| 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.
Every verdict on record (6)
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.