Derivative of \( \displaystyle \left(\frac{1}{4} - x\right) e^{4 x} \)
Problem 2.632 · medium
Differentiate \( \displaystyle f(x) = \left(\frac{1}{4} - x\right) e^{4 x} \).
- \[ \frac{d}{d x} \left(\frac{1}{4} - x\right) e^{4 x} \]Start with the derivative of the function.✓ Proved
- \[ = \left(\frac{1}{4} - x\right) \frac{d}{d x} e^{4 x} + e^{4 x} \frac{d}{d x} \left(\frac{1}{4} - x\right) \]productApply the product rule.✓ Proved
- \[ = \left(\frac{1}{4} - x\right) \frac{d}{d x} e^{4 x} - e^{4 x} \]derivative algebraDifferentiate the first part of the product. Simplify the first term.✓ Proved
- \[ = 4 \left(\frac{1}{4} - x\right) e^{4 x} - e^{4 x} \]derivativeDifferentiate the second part of the product using the chain rule.✓ Proved
- \[ = \left(1 - 4 x\right) e^{4 x} - e^{4 x} \]algebraDistribute the 4 into the parentheses.✓ Proved
- \[ = - 4 x e^{4 x} \]algebra simplifyFactor out the common exponential term. Combine like terms.✓ Proved
Answer \( - 4 x e^{4 x} \)
✓ 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.
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 |
| 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: passqwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 5 applies the chain rule but is labeled 'derivative'; while 'derivative' is acceptable for unfolding d/dx, the note explicitly mentions the chain rule, creating a mismatch between the label and the explanation. More critically, Step 3 applies the derivative to a sum (1/4 - x) but is labeled 'derivative' which is acceptable, however, Step 5's label 'derivative' is technically correct for the outer function, but the step combines the chain rule application. The main issue is that Step 5's note says 'using the chain rule' but the label is 'derivative'. In this strict vocabulary, 'chain' is a valid label. Using 'derivative' for a composite function differentiation is often considered a defect if 'chain' is available and more precise, but the prompt says 'derivative' means unfolding d/dx on a known form. exp(4x) is not a 'known form' in the base sense (like exp(x)), it requires chain rule. Therefore, labeling a chain rule step as 'derivative' is a defect because 'chain' is in the vocabulary and is the correct label for differentiating a composite function. The label 'derivative' is reserved for base functions like exp(x), sin(x), etc.gpt-oss:20b: pass 2026-09-21
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.