Home›Calculus 1›Product rule›Problem 2.660

Derivative of \( \displaystyle \left(x + \frac{1}{3}\right) e^{3 x + 2} \)

Problem 2.660 · medium Video

Differentiate \( \displaystyle f(x) = \left(x + \frac{1}{3}\right) e^{3 x + 2} \).

\[ \frac{d}{d x} \left(x + \frac{1}{3}\right) e^{3 x + 2} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} \left(x e^{3 x + 2} + \frac{e^{3 x + 2}}{3}\right) \]
sumDistribute the first term.✓ Proved
\[ = \frac{d}{d x} x e^{3 x + 2} + \frac{d}{d x} \frac{e^{3 x + 2}}{3} \]
sumApply the sum rule.✓ Proved
\[ = x \frac{d}{d x} e^{3 x + 2} + e^{3 x + 2} \frac{d}{d x} x + \frac{d}{d x} \frac{e^{3 x + 2}}{3} \]
productApply the product rule to the first term.✓ Proved
\[ = x \frac{d}{d x} e^{3 x + 2} + e^{3 x + 2} + \frac{d}{d x} \frac{e^{3 x + 2}}{3} \]
derivativeDifferentiate x.✓ Proved
\[ = x \frac{d}{d x} e^{3 x + 2} + e^{3 x + 2} + \frac{\frac{d}{d x} e^{3 x + 2}}{3} \]
constant-multipleFactor out the constant 1/3.✓ Proved
\[ = 3 x e^{3 x + 2} + 2 e^{3 x + 2} \]
chain algebra algebraApply the chain rule to the exponential terms. Simplify the constant multiplication. Combine like terms.✓ Proved
\[ = \left(3 x + 2\right) e^{3 x + 2} \]
simplifyFactor out the common exponential term.✓ Proved

Answer \( \left(3 x + 2\right) e^{3 x + 2} \)

Top: the graph of f = (x + 1/3)·e^(3x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3(x + 1/3)·e^(3x + 2) + e^(3x + 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. 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
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
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: pass
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-11: fail — The math is correct: f' = e^(3x+2) + 3(x+1/3)e^(3x+2) = (3x+2)e^(3x+2), and every line was proved. Line 2 only distributes the product, which is 'algebra' or 'rewrite', but it is labeled 'sum'. The sum rule is not applied until line 3, so this is a mislabel but does not affect correctness.

Every verdict on record (7)

claude-sonnet-5-5: fail (style) 2026-10-11 — The math is correct: f' = e^(3x+2) + 3(x+1/3)e^(3x+2) = (3x+2)e^(3x+2), and every line was proved. Line 2 only distributes the product, which is 'algebra' or 'rewrite', but it is labeled 'sum'. The sum rule is not applied until line 3, so this is a mislabel but does not affect correctness.
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels accurately describe the operations performed.
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies differentiation rules step-by-step, adhering to the single-change constraint and using valid labels from the fixed vocabulary.
gpt-oss:20b: fail (error) 2026-09-21 — Step 7 applies the chain rule to two separate terms in a single step, violating the rule that each step must change only one thing. The label "chain" is also misleading because it covers two distinct applications.

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.