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

Problem 2.612 · hard

1. \[ \frac{d}{d x} \left(2 x - 1\right)^{2} e^{1 - 2 x} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \left(2 x - 1\right)^{2} \frac{d}{d x} e^{1 - 2 x} + e^{1 - 2 x} \frac{d}{d x} \left(2 x - 1\right)^{2} \]
   productApply the product rule.✓ Proved
3. \[ = \left(2 x - 1\right)^{2} \frac{d}{d x} e^{1 - 2 x} + \left(4 x - 2\right) e^{1 - 2 x} \frac{d}{d x} \left(2 x - 1\right) \]
   powerDifferentiate the first part using the power rule.✓ Proved
4. \[ = \left(2 x - 1\right)^{2} \frac{d}{d x} e^{1 - 2 x} + \left(8 x - 4\right) e^{1 - 2 x} \]
   derivative algebraDifferentiate the inner part of the power term. Simplify the first term.✓ Proved
5. \[ = \left(2 x - 1\right)^{2} e^{1 - 2 x} \frac{d}{d x} \left(1 - 2 x\right) + \left(8 x - 4\right) e^{1 - 2 x} \]
   chainApply the chain rule to the exponential term.✓ Proved
6. \[ = - 2 \left(2 x - 1\right)^{2} e^{1 - 2 x} + \left(8 x - 4\right) e^{1 - 2 x} \]
   derivative algebraDifferentiate the exponent. Simplify the expression.✓ Proved
7. \[ = \left(6 - 4 x\right) \left(2 x - 1\right) e^{1 - 2 x} \]
   algebra algebra simplifyFactor out common terms. Distribute the -2 inside the parentheses. Combine like terms.✓ Proved