Home›Calculus 1›Chain rule›Problem 2.166

Derivative of \( \displaystyle \left(\frac{1}{2} - x\right) e^{2 x} \)

Problem 2.166 · hard Video

Differentiate \( \displaystyle f(x) = \left(\frac{1}{2} - x\right) e^{2 x} \).

\[ \frac{d}{d x} \left(\frac{1}{2} - x\right) e^{2 x} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} \left(- x e^{2 x} + \frac{e^{2 x}}{2}\right) \]
algebraDistribute the term.✓ Proved
\[ = - \frac{d}{d x} x e^{2 x} + \frac{d}{d x} \frac{e^{2 x}}{2} \]
sumApply the difference rule.✓ Proved
\[ = - \frac{d}{d x} x e^{2 x} + \frac{\frac{d}{d x} e^{2 x}}{2} \]
constant-multipleFactor out the constant.✓ Proved
\[ = - x \frac{d}{d x} e^{2 x} - e^{2 x} \frac{d}{d x} x + \frac{\frac{d}{d x} e^{2 x}}{2} \]
productApply the product rule to the second term.✓ Proved
\[ = - 2 x e^{2 x} \]
chain simplify algebra simplifyApply the chain rule to the exponential terms. Simplify the coefficients. Distribute the negative sign. Combine like terms.✓ Proved

Answer \( - 2 x e^{2 x} \)

Top: the graph of f = (1/2 - x)·e^(2x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2(1/2 - x)·e^(2x) - e^(2x).

✓ 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
9	✓ 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
deepseek-r1:70b: pass
qwen3.6:27b-mlx: fail (error) — Step 6 applies the chain rule to both exponential terms simultaneously, violating the single-rule-per-step constraint. Additionally, the label 'chain' is incorrect for the differentiation of x (Derivative(x, x) -> 1), which is a basic derivative, not a chain rule application.

Senior review claude-sonnet-5-5, 2026-10-05: pass — The derivation is correct: product rule, then chain rule on exp(2x), then cancellation to -2*x*exp(2*x), which matches the stated answer. The labels are legitimate, and step 6 is a single substitution of derivatives.

qwen3.6:27b-mlx: dismiss — Both occurrences of Derivative(exp(2*x),x) are the same chain-rule evaluation, so replacing them counts as one change; Derivative(x,x)->1 is the trivial derivative of a simple function and is acceptably folded into that step; the 'chain' label names a rule actually applied.

Every verdict on record (16)

claude-sonnet-5-5: pass 2026-10-05 — The derivation is correct: product rule, then chain rule on exp(2x), then cancellation to -2*x*exp(2*x), which matches the stated answer. The labels are legitimate, and step 6 is a single substitution of derivatives.
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 6 applies the chain rule to both exponential terms simultaneously, violating the single-rule-per-step constraint. Additionally, the label 'chain' is incorrect for the differentiation of x (Derivative(x, x) -> 1), which is a basic derivative, not a chain rule application.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 6 applies the chain rule to both exponential terms simultaneously, violating the one-change-per-step constraint. It should be split into two steps.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: fail (error) 2026-09-18 — Step 6 applies the chain rule to both exponential terms simultaneously, violating the one-rule-per-step constraint. Additionally, the label 'chain' is insufficient because the step also evaluates the derivatives of the inner functions (e.g., d/dx(2x) = 2) and the outer exponential, which are distinct differentiation rules.
deepseek-r1:70b: pass 2026-09-18
gpt-oss:20b: fail (error) 2026-09-18 — Step 6 applies the chain rule twice in a single line, changing two terms at once. Each step should modify only one part of the expression.
gpt-oss:20b: fail 2026-09-17 — Step 3 incorrectly labels the rule as "sum" while actually applying the difference rule, which misleads the student about the rule being used.
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.