Derivative of \( \displaystyle e^{\left(2 x - 1\right)^{2} + 1} \)
Problem 2.616 · hard Video
Differentiate \( \displaystyle f(x) = e^{\left(2 x - 1\right)^{2} + 1} \).
- \[ \frac{d}{d x} e^{\left(2 x - 1\right)^{2} + 1} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = e^{\left(2 x - 1\right)^{2} + 1} \frac{d}{d x} \left(\left(2 x - 1\right)^{2} + 1\right) \]chainApply the chain rule.✓ Proved
- \[ = \left(\frac{d}{d x} 1 + \frac{d}{d x} \left(2 x - 1\right)^{2}\right) e^{\left(2 x - 1\right)^{2} + 1} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = e^{\left(2 x - 1\right)^{2} + 1} \frac{d}{d x} \left(2 x - 1\right)^{2} \]constant algebraThe derivative of a constant is zero. Simplify the expression by removing the zero term.✓ Proved
- \[ = \left(4 x - 2\right) e^{\left(2 x - 1\right)^{2} + 1} \frac{d}{d x} \left(2 x - 1\right) \]powerApply the power rule and chain rule to the squared term.✓ Proved
- \[ = \left(8 x - 4\right) e^{\left(2 x - 1\right)^{2} + 1} \]derivative algebra simplifyDifferentiate the linear term inside. Multiply the constants. Distribute and simplify the final expression.✓ Proved
Answer \( \left(8 x - 4\right) e^{\left(2 x - 1\right)^{2} + 1} \)
✓ 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: passqwen3.6:27b-mlx: fail (error) — Step 6 applies both the power rule and the chain rule simultaneously, violating the constraint that each step must change only one thing. It should be split into a step applying the power rule to get 2*(2*x-1)*Derivative(2*x-1, x) is incorrect because the power rule alone on (u)^2 gives 2*u*u', but here the derivative operator is still present on the inner term, implying the chain rule was also applied. Specifically, the transition from Derivative((2*x-1)^2, x) to 2*(2*x-1)*Derivative(2*x-1, x) requires the chain rule. The label 'power' is insufficient/inaccurate for a step that also invokes the chain rule structure, or rather, the step performs two logical differentiation rules at once.
Senior review claude-sonnet-5-5, 2026-10-11: pass — All steps are correct and each one changes a single thing. SymPy proved every line, and the labels are acceptable. Step 6 is a power rule with its chain factor, and step 7 evaluates the inner derivative.
qwen3.6:27b-mlx: dismiss — Differentiating (2x-1)^2 as 2*(2x-1)*d/dx(2x-1) is the power rule in its general (chain-form) version, which the contract treats as one step. The label 'power' names a rule that was legitimately applied, and the inner derivative is left unevaluated, so only one change is made.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-11 — All steps are correct and each one changes a single thing. SymPy proved every line, and the labels are acceptable. Step 6 is a power rule with its chain factor, and step 7 evaluates the inner derivative.qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 6 applies both the power rule and the chain rule simultaneously, violating the constraint that each step must change only one thing. It should be split into a step applying the power rule to get 2*(2*x-1)*Derivative(2*x-1, x) is incorrect because the power rule alone on (u)^2 gives 2*u*u', but here the derivative operator is still present on the inner term, implying the chain rule was also applied. Specifically, the transition from Derivative((2*x-1)^2, x) to 2*(2*x-1)*Derivative(2*x-1, x) requires the chain rule. The label 'power' is insufficient/inaccurate for a step that also invokes the chain rule structure, or rather, the step performs two logical differentiation rules at once.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 6 applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. It should be split into a power rule step and a subsequent chain rule step.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 6 applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. It should be split into a power rule step followed by a chain rule step.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.