Derivative of \( \displaystyle \left(- x - \frac{1}{2}\right) e^{2 x + 2} \)
Problem 2.884 · medium Video
Differentiate \( \displaystyle f(x) = \left(- x - \frac{1}{2}\right) e^{2 x + 2} \).
- \[ \frac{d}{d x} \left(- x - \frac{1}{2}\right) e^{2 x + 2} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \left(- x - \frac{1}{2}\right) \frac{d}{d x} e^{2 x + 2} + e^{2 x + 2} \frac{d}{d x} \left(- x - \frac{1}{2}\right) \]productApply the product rule.✓ Proved
- \[ = \left(- x - \frac{1}{2}\right) \frac{d}{d x} e^{2 x + 2} + e^{2 x + 2} \frac{d}{d x} \left(- \frac{1}{2}\right) + e^{2 x + 2} \frac{d}{d x} \left(- x\right) \]sumApply the sum rule to the first term.✓ Proved
- \[ = \left(- x - \frac{1}{2}\right) \frac{d}{d x} e^{2 x + 2} - e^{2 x + 2} \]derivativeDifferentiate the individual terms.✓ Proved
- \[ = - \left(x + \frac{1}{2}\right) \frac{d}{d x} e^{2 x + 2} - e^{2 x + 2} \]algebraSimplify the expression.✓ Proved
- \[ = - \left(x + \frac{1}{2}\right) e^{2 x + 2} \frac{d}{d x} \left(2 x + 2\right) - e^{2 x + 2} \]exponentialDifferentiate the exponential term using the chain rule.✓ Proved
- \[ = - 2 \left(x + \frac{1}{2}\right) e^{2 x + 2} - e^{2 x + 2} \]derivativeDifferentiate the exponent.✓ Proved
- \[ = - \left(2 x + 1\right) e^{2 x + 2} - e^{2 x + 2} \]algebraDistribute the constant and simplify.✓ Proved
- \[ = - 2 x e^{2 x + 2} - 2 e^{2 x + 2} \]algebra algebraDistribute the term. Combine like terms.✓ Proved
- \[ = \left(- 2 x - 2\right) e^{2 x + 2} \]simplifyFactor out the common term.✓ Proved
Answer \( 2 \left(- x - 1\right) e^{2 x + 2} \)
✓ 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 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 11 | ✓ 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: fail (error) — Step 5 incorrectly changes the sign of the term `(-x - 1/2)` to `-(x + 1/2)` while simultaneously dropping the derivative operator `Derivative(..., x)`. This step applies algebraic simplification and differentiation (or rather, fails to differentiate) in a single step, violating the one-rule-per-step constraint and resulting in a mathematically incorrect transition.
Senior review claude-sonnet-5-5, 2026-10-13: pass — Every line was proved equivalent, the final answer matches, and the rule labels are legitimate. Line 11 is labeled 'simplify' where 'algebra' would also fit, which is acceptable.
qwen3.6:27b-mlx: dismiss — Line 5 still contains Derivative(exp(2*x + 2), x), so no derivative operator was dropped. Rewriting (-x - 1/2) as -(x + 1/2) and cleaning up -1*exp and 0*exp are all algebra, and SymPy proved the equivalence.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-13 — Every line was proved equivalent, the final answer matches, and the rule labels are legitimate. Line 11 is labeled 'simplify' where 'algebra' would also fit, which is acceptable.qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 5 incorrectly changes the sign of the term `(-x - 1/2)` to `-(x + 1/2)` while simultaneously dropping the derivative operator `Derivative(..., x)`. This step applies algebraic simplification and differentiation (or rather, fails to differentiate) in a single step, violating the one-rule-per-step constraint and resulting in a mathematically incorrect transition.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 5 incorrectly changes the sign of the third term from positive to negative (from `+ (-x - 1/2)*...` to `- (x + 1/2)*...`) without justification, effectively applying an algebraic simplification to one term while dropping the sign change logic or misapplying it. Additionally, Step 6 labels the application of the chain rule as 'exponential', which is not in the allowed vocabulary for that specific structural change (though 'chain' is allowed, the step also performs the differentiation of the inner function in step 7, making step 6 a partial application that is confusingly labeled).gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 5 incorrectly changes the sign of the term `(-x - 1/2)` to `-(x + 1/2)` while simultaneously dropping the `Derivative` operator, applying two distinct operations (algebraic simplification and differentiation) in a single step. Additionally, Step 6 is labeled 'exponential' but describes using the chain rule, which is a mislabeling defect.gpt-oss:20b: pass 2026-09-26
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 2026-09-27 with SymPy 1.14.0.