Derivative of \( \displaystyle \left(5 x + \frac{5}{4}\right) e^{4 x + 2} \)
Problem 2.1427 · medium
Differentiate \( \displaystyle f(x) = \left(5 x + \frac{5}{4}\right) e^{4 x + 2} \).
- \[ \frac{d}{d x} \left(5 x + \frac{5}{4}\right) e^{4 x + 2} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(5 x e^{4 x + 2} + \frac{5 e^{4 x + 2}}{4}\right) \]sumDistribute the terms.✓ Proved
- \[ = \frac{d}{d x} 5 x e^{4 x + 2} + \frac{d}{d x} \frac{5 e^{4 x + 2}}{4} \]sumApply the sum rule.✓ Proved
- \[ = 5 \frac{d}{d x} x e^{4 x + 2} + \frac{5 \frac{d}{d x} e^{4 x + 2}}{4} \]constant-multipleFactor out the constants.✓ Proved
- \[ = 5 x \frac{d}{d x} e^{4 x + 2} + 5 e^{4 x + 2} \frac{d}{d x} x + \frac{5 \frac{d}{d x} e^{4 x + 2}}{4} \]productApply the product rule to the first term.✓ Proved
- \[ = 5 x \frac{d}{d x} e^{4 x + 2} + 5 e^{4 x + 2} + \frac{5 \frac{d}{d x} e^{4 x + 2}}{4} \]derivativeDifferentiate x.✓ Proved
- \[ = 5 x e^{4 x + 2} \frac{d}{d x} \left(4 x + 2\right) + 5 e^{4 x + 2} + \frac{5 \frac{d}{d x} e^{4 x + 2}}{4} \]chainPreparing for chain rule. Apply the chain rule to the exponential term.✓ Proved
- \[ = 20 x e^{4 x + 2} + 10 e^{4 x + 2} \]derivative algebra algebra simplifyDifferentiate the exponent 4*x + 2. Simplify the terms. Distribute the 5. Combine like terms.✓ Proved
Answer \( \left(20 x + 10\right) e^{4 x + 2} \)
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 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 12 | ✓ 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 7 applies two rules at once (incorrectly rewrites the derivative of exp and introduces a division by (4*x+2) that is later cancelled). The chain rule is misapplied in step 8, and the intermediate expression in step 7 is mathematically incorrect.qwen3.6:27b-mlx: fail (error) — Step 7 introduces a nonsensical expression involving division by (4*x + 2) and multiplication by (4*x + 2) that does not follow from the previous step and is not a valid application of any rule. Step 8 then claims to apply the chain rule, but the transition from Step 7 to Step 8 relies on cancelling the erroneous terms introduced in Step 7 rather than a correct derivation.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 7 introduces a nonsensical expression involving division by (4*x + 2) and multiplication by (4*x + 2) that does not follow from the previous step and is not a valid application of any rule. Step 8 then claims to apply the chain rule, but the transition from Step 7 to Step 8 relies on cancelling the erroneous terms introduced in Step 7 rather than a correct derivation.gpt-oss:20b: fail (error) 2026-10-03 — Step 7 applies two rules at once (incorrectly rewrites the derivative of exp and introduces a division by (4*x+2) that is later cancelled). The chain rule is misapplied in step 8, and the intermediate expression in step 7 is mathematically incorrect.qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 7 introduces a nonsensical expression involving division by (4*x + 2) and multiplication by (4*x + 2) that does not follow from the previous step and is not a valid application of the chain rule. Step 8 then claims to apply the chain rule, but the transition from Step 7 to Step 8 is algebraically messy and the label 'chain' is misapplied to a simplification of the erroneous Step 7 rather than a direct differentiation.gpt-oss:20b: fail (error) 2026-10-03 — Step 7 applies an incorrect manipulation of the derivative of the exponential term, introducing an unnecessary division by (4*x+2) and then multiplying by it again. This step combines two distinct operations (a faulty chain‑rule application and an algebraic simplification) in one line and does not correctly represent the derivative of exp(4*x+2).
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.