Derivative of \( \displaystyle \left(2 x + \frac{2}{5}\right) e^{5 x + 2} \)
Problem 2.148 · hard
Differentiate \( \displaystyle f(x) = \left(2 x + \frac{2}{5}\right) e^{5 x + 2} \).
- \[ \frac{d}{d x} \left(2 x + \frac{2}{5}\right) e^{5 x + 2} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \left(2 x + \frac{2}{5}\right) \frac{d}{d x} e^{5 x + 2} + e^{5 x + 2} \frac{d}{d x} \frac{2}{5} + e^{5 x + 2} \frac{d}{d x} 2 x \]productApply the product rule for three terms.✓ Proved
- \[ = \left(2 x + \frac{2}{5}\right) \frac{d}{d x} e^{5 x + 2} + e^{5 x + 2} \frac{d}{d x} \frac{2}{5} + 2 e^{5 x + 2} \]derivativeDifferentiate the first term.✓ Proved
- \[ = \left(2 x + \frac{2}{5}\right) \frac{d}{d x} e^{5 x + 2} + 2 e^{5 x + 2} \]derivative simplifyDifferentiate the second term. Simplify the zero term.✓ Proved
- \[ = \left(2 x + \frac{2}{5}\right) e^{5 x + 2} \frac{d}{d x} \left(5 x + 2\right) + 2 e^{5 x + 2} \]chainApply the chain rule to the exponential term.✓ Proved
- \[ = 5 \left(2 x + \frac{2}{5}\right) e^{5 x + 2} + 2 e^{5 x + 2} \]derivativeDifferentiate the exponent.✓ Proved
- \[ = \left(10 x + 2\right) e^{5 x + 2} + 2 e^{5 x + 2} \]algebraDistribute the 5 into the parentheses.✓ Proved
- \[ = \left(10 x + 4\right) e^{5 x + 2} \]algebra simplifyFactor out the common exponential term. Combine the constant terms.✓ Proved
Answer \( \left(10 x + 4\right) e^{5 x + 2} \)
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. A reviewer's objection to the explanation stands; see 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 |
| 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 twice at once by splitting the first factor into two terms; each product rule should be a separate step. The labeling is correct, but the step violates the one‑change‑per‑step rule.deepseek-r1:70b: fail (error) — Step 2 misapplies the product rule by incorrectly expanding it to three terms instead of two, which is a defect and misleading.qwen3.6:27b-mlx: fail (error) — Step 2 incorrectly applies the product rule to a product of two factors as if it were a sum of three terms, introducing a spurious term involving the derivative of the constant 2/5. The product rule for u*v is u'v + uv', not u'v + c'v + uv' where c is part of u.
Every verdict on record (15)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 2 incorrectly applies the product rule to a product of two factors as if it were a sum of three terms, introducing a spurious term involving the derivative of the constant 2/5. The product rule for u*v is u'v + uv', not u'v + c'v + uv' where c is part of u.gpt-oss:20b: fail (error) 2026-09-20 — Step 2 applies the product rule twice at once by splitting the first factor into two terms; each product rule should be a separate step. The labeling is correct, but the step violates the one‑change‑per‑step rule.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 2 incorrectly applies the product rule to a product of two factors as if it were a sum of three terms, resulting in an invalid expansion. The product rule for u*v is u'v + uv', not u'v + c'v + uv' where c is a constant split from u.gpt-oss:20b: fail (error) 2026-09-20 — Step 2 incorrectly applies the product rule to three terms; the product rule only applies to a product of two functions, so the expansion into three separate derivative terms is mathematically invalid.qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 2 incorrectly applies the product rule to a product of two terms as if it were a product of three terms, splitting the first factor (2*x + 2/5) into two separate terms while keeping the second factor intact. This violates the 'one rule per step' constraint and misapplies the product rule structure.deepseek-r1:70b: fail (error) 2026-09-19 — Step 2 misapplies the product rule by incorrectly expanding it to three terms instead of two, which is a defect and misleading.gpt-oss:20b: fail (error) 2026-09-19 — Step 2 incorrectly applies the product rule to the sum (2*x + 2/5) as if it were three separate factors. The correct product rule would produce only two terms: Derivative(2*x + 2/5, x)*exp(5*x + 2) + (2*x + 2/5)*Derivative(exp(5*x + 2), x).qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 2 incorrectly applies the product rule to a product of two terms as if it were a sum of three terms, introducing a spurious derivative of the constant 2/5 as a separate factor. The product rule for f*g is f'*g + f*g', not f'*g + c'*g + f*g' where c is part of f.deepseek-r1:70b: fail (misleading) 2026-09-19 — Step 2 incorrectly applies the product rule for three terms instead of two.gpt-oss:20b: fail (error) 2026-09-19 — Step 2 incorrectly applies the product rule to three terms instead of two, treating the expression as a product of three factors. The correct product rule should only involve the two factors (2*x+2/5) and exp(5*x+2).qwen3.6:27b-mlx: fail (error) 2026-09-18 — Step 2 incorrectly applies the product rule to a product of two terms as if it were a product of three terms, introducing a spurious term involving the derivative of the constant 2/5. The product rule for u*v is u'v + uv', not u'v + c'v + uv'.deepseek-r1:70b: fail (misleading) 2026-09-18 — Step 2 incorrectly states that the product rule applies to three terms, which is misleading as the product rule is for two functions.gpt-oss:20b: fail (error) 2026-09-18 — Step 2 incorrectly applies the product rule to three terms. The product rule only applies to two factors; the correct application would be Derivative(2*x+2/5, x)*exp(5*x+2) + (2*x+2/5)*Derivative(exp(5*x+2), x).gpt-oss:20b: fail 2026-09-17 — Step 2 incorrectly labels the operation as a single product rule while actually applying both the sum rule (to split 2*x+2/5) and the product rule at once. The label "product" does not match the combined rules used, which could mislead a student.deepseek-r1:70b: pass 2026-09-17
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.