Home›Calculus 1›Product rule›Problem 2.551

Derivative of \( \displaystyle \left(\frac{1}{3} - x\right) e^{3 x} \)

Problem 2.551 · medium Video

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 \( - 3 x e^{3 x} \)

Top: the graph of f = (1/3 - x)·e^(3x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3(1/3 - x)·e^(3x) - e^(3x).

✓ 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
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 (error) — Step 3 applies two derivative rules in a single line (derivative of exp(3*x) and derivative of (1/3 - x)). Each rule application must be a separate step.
qwen3.6:27b-mlx: pass — The solution correctly applies the product rule, chain rule (embedded in the derivative step), and algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.

Senior review claude-sonnet-5-5, 2026-10-10: pass — The math is correct, every line was proved equivalent, and the final answer -3*x*exp(3*x) is right. The rule labels fit: product, derivative, algebra, simplify. Line 3 evaluates both derivatives in one step, which is acceptable.

gpt-oss:20b: dismiss — Evaluating the two derivatives in the product-rule result is one 'derivative' step, and the contract treats applying a rule to several terms (with simple inner derivatives done as part of the chain step) as one change; SymPy also verified the equality.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-10 — The math is correct, every line was proved equivalent, and the final answer -3*x*exp(3*x) is right. The rule labels fit: product, derivative, algebra, simplify. Line 3 evaluates both derivatives in one step, which is acceptable.
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the product rule, chain rule (embedded in the derivative step), and algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: fail (error) 2026-09-26 — Step 3 applies two derivative rules in a single line (derivative of exp(3*x) and derivative of (1/3 - x)). Each rule application must be a separate step.
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 2026-09-21
qwen3.6:27b-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.