Home›Calculus 1›Product rule›Problem 2.632

Derivative of \( \displaystyle \left(\frac{1}{4} - x\right) e^{4 x} \)

Problem 2.632 · medium Video

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 \( - 4 x e^{4 x} \)

Top: the graph of f = (1/4 - x)·e^(4x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 4(1/4 - x)·e^(4x) - e^(4x).

✓ 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. How we verify

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: pass
qwen3.6:27b-mlx: fail (style) — Step 5 is labeled 'derivative' but the note explicitly mentions using the 'chain rule'. While the label 'derivative' is technically acceptable for unfolding d/dx(exp(u)), the step combines the chain rule application with the derivative lookup, which is acceptable, but the label 'derivative' is less precise than 'chain' if 'chain' were available. However, 'chain' IS in the vocabulary. The step applies the chain rule to exp(4x). The label 'derivative' usually refers to basic derivatives (like d/dx sin x). Applying the chain rule is a distinct rule in the vocabulary. Therefore, labeling a chain rule step as just 'derivative' is a defect because a more specific and correct label ('chain') exists in the fixed vocabulary.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 5 is labeled 'derivative' but the note explicitly mentions using the 'chain rule'. While the label 'derivative' is technically acceptable for unfolding d/dx(exp(u)), the step combines the chain rule application with the derivative lookup, which is acceptable, but the label 'derivative' is less precise than 'chain' if 'chain' were available. However, 'chain' IS in the vocabulary. The step applies the chain rule to exp(4x). The label 'derivative' usually refers to basic derivatives (like d/dx sin x). Applying the chain rule is a distinct rule in the vocabulary. Therefore, labeling a chain rule step as just 'derivative' is a defect because a more specific and correct label ('chain') exists in the fixed vocabulary.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-21
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-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.