Derivative of \( \displaystyle \left(3 x + \frac{3}{5}\right) e^{5 x + 2} \)
Problem 2.1321 · medium
Differentiate \( \displaystyle f(x) = \left(3 x + \frac{3}{5}\right) e^{5 x + 2} \).
- \[ \frac{d}{d x} \left(3 x + \frac{3}{5}\right) e^{5 x + 2} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \left(3 x + \frac{3}{5}\right) \frac{d}{d x} e^{5 x + 2} + e^{5 x + 2} \frac{d}{d x} \frac{3}{5} + e^{5 x + 2} \frac{d}{d x} 3 x \]productApply the product rule.✓ Proved
- \[ = \left(3 x + \frac{3}{5}\right) \frac{d}{d x} e^{5 x + 2} + e^{5 x + 2} \frac{d}{d x} \frac{3}{5} + 3 e^{5 x + 2} \]derivativeDifferentiate the first term.✓ Proved
- \[ = \left(3 x + \frac{3}{5}\right) \frac{d}{d x} e^{5 x + 2} + 3 e^{5 x + 2} \]derivative algebraDifferentiate the second term. Simplify the zero term.✓ Proved
- \[ = \left(3 x + \frac{3}{5}\right) e^{5 x + 2} \frac{d}{d x} \left(5 x + 2\right) + 3 e^{5 x + 2} \]chainApply the chain rule to the exponential term.✓ Proved
- \[ = 5 \left(3 x + \frac{3}{5}\right) e^{5 x + 2} + 3 e^{5 x + 2} \]derivativeDifferentiate the exponent.✓ Proved
- \[ = \left(15 x + 3\right) e^{5 x + 2} + 3 e^{5 x + 2} \]algebraDistribute the 5.✓ Proved
- \[ = 15 x e^{5 x + 2} + 6 e^{5 x + 2} \]algebra algebraDistribute the exponential term. Combine like terms.✓ Proved
- \[ = \left(15 x + 6\right) e^{5 x + 2} \]simplifyFactor out the common exponential term.✓ Proved
Answer \( \left(15 x + 6\right) e^{5 x + 2} \)
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 |
| 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 |
| 11 | ✓ 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 2 applies the product rule to each summand of (3*x+3/5) in one go, effectively combining the sum rule and product rule simultaneously, which violates the rule that each step must change only one thing.qwen3.6:27b-mlx: pass — The solution correctly applies the product rule, chain rule, and algebraic simplifications. 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 2026-10-03 — The solution correctly applies the product rule, chain rule, and algebraic simplifications. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-10-03 — Step 2 applies the product rule to each summand of (3*x+3/5) in one go, effectively combining the sum rule and product rule simultaneously, which violates the rule that each step must change only one thing.qwen3.6:27b-mlx: fail (error) 2026-09-30 — Step 2 incorrectly applies the product rule to a sum of two terms (3x + 3/5) as if it were a single factor, resulting in three terms instead of the correct two. The product rule (uv)' = u'v + uv' was applied to u = 3x + 3/5 and v = exp(5x+2), but the step expanded u' as if u were two separate factors multiplied by v, violating the 'one rule per step' constraint and the definition of the product rule.gpt-oss:20b: fail (error) 2026-09-30 — Step 2 applies both the product rule and linearity of differentiation in one line, violating the one‑rule‑per‑step rule. The correct approach would first differentiate the sum (3*x + 3/5) and then apply the product rule, or vice versa, in separate steps.
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 2026-10-03 with SymPy 1.14.0.