Derivative of \( \displaystyle \left(5 x + 1\right) e^{5 x + 2} \)
Problem 2.149 · hard
Differentiate \( \displaystyle f(x) = \left(5 x + 1\right) e^{5 x + 2} \).
- \[ \frac{d}{d x} \left(5 x + 1\right) e^{5 x + 2} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} 5 x e^{5 x + 2} + \frac{d}{d x} e^{5 x + 2} \]sum constantApply the sum rule. Simplify the constant term.✓ Proved
- \[ = 5 x \frac{d}{d x} e^{5 x + 2} + e^{5 x + 2} \frac{d}{d x} 5 x + \frac{d}{d x} e^{5 x + 2} \]productApply the product rule to the first term.✓ Proved
- \[ = 5 x \frac{d}{d x} e^{5 x + 2} + 5 e^{5 x + 2} + \frac{d}{d x} e^{5 x + 2} \]derivativeDifferentiate 5*x.✓ Proved
- \[ = 5 x e^{5 x + 2} \frac{d}{d x} \left(5 x + 2\right) + e^{5 x + 2} \frac{d}{d x} \left(5 x + 2\right) + 5 e^{5 x + 2} \]productApply the chain rule to the exponential terms.✓ Proved
- \[ = 25 x e^{5 x + 2} + 10 e^{5 x + 2} \]derivative algebra algebraDifferentiate 5*x + 2. Multiply the terms. Combine like terms.✓ Proved
- \[ = \left(25 x + 10\right) e^{5 x + 2} \]algebraFactor out the common exponential term.✓ Proved
Answer \( \left(25 x + 10\right) e^{5 x + 2} \)
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 |
| 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: passdeepseek-r1:70b: fail (error) — Step 6 incorrectly labels the application of the chain rule as 'product'.qwen3.6:27b-mlx: fail (error) — Step 6 is labeled 'product' but applies the chain rule to the exponential terms. The label must match the rule applied; 'chain' is the correct label from the vocabulary.
Every verdict on record (15)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 6 is labeled 'product' but applies the chain rule to the exponential terms. The label must match the rule applied; 'chain' is the correct label from the vocabulary.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 6 is labeled 'product' but applies the chain rule to the exponential terms. Additionally, Step 3 is labeled 'constant' but performs algebraic simplification (1*exp -> exp), which should be labeled 'algebra' or 'simplify'.gpt-oss:20b: fail (error) 2026-09-20 — Step 6 applies the chain rule to both exponential terms in a single step and labels it as "product", violating the rule‑granularity and naming requirements.qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 6 is labeled 'product' but applies the chain rule to differentiate the exponential function; the label does not match the rule applied.deepseek-r1:70b: fail (error) 2026-09-19 — Step 6 incorrectly labels the application of the chain rule as 'product'.gpt-oss:20b: fail (style) 2026-09-19 — Step 6 incorrectly labels the application of the chain rule as "product". The rule used is the chain rule, not the product rule.qwen3.6:27b-mlx: fail (misleading) 2026-09-19 — Step 6 is labeled 'product' but applies the chain rule to the exponential terms; the label does not match the rule applied. Additionally, Step 2 splits the product into a sum, which is a valid algebraic rewrite, but Step 6's mislabeling is a clear defect.deepseek-r1:70b: fail (misleading) 2026-09-19 — Step 6 incorrectly labels the application of the chain rule as 'product'. This could mislead a student into thinking the product rule was applied instead of the chain rule.gpt-oss:20b: fail (error) 2026-09-19 — Step 2 incorrectly applies the sum rule to the product (5*x+1)*exp(5*x+2). The derivative of a product cannot be split into the sum of the derivatives of its factors; the product rule must be used instead.qwen3.6:27b-mlx: fail (misleading) 2026-09-18 — Step 6 is labeled 'product' but applies the chain rule to the exponential terms; the label does not match the rule applied. Additionally, Step 6 combines the differentiation of two separate terms from Step 5 into one step, violating the one-change-per-step constraint.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: fail (style) 2026-09-18 — Step 6 incorrectly labels the application of the chain rule as "product". The correct label should be "chain".gpt-oss:20b: fail 2026-09-17 — Step 2 incorrectly labels the distributive property as a sum rule and the note says "Apply the sum rule" which misleads the student. Step 6 labels the operation as a product rule while the note correctly describes applying the chain rule to the exponential; the rule label is therefore mismatched.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.