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

Problem 2.166 · hard

1. \[ \frac{d}{d x} \left(\frac{1}{2} - x\right) e^{2 x} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- x e^{2 x} + \frac{e^{2 x}}{2}\right) \]
   algebraDistribute the term.✓ Proved
3. \[ = - \frac{d}{d x} x e^{2 x} + \frac{d}{d x} \frac{e^{2 x}}{2} \]
   sumApply the difference rule.✓ Proved
4. \[ = - \frac{d}{d x} x e^{2 x} + \frac{\frac{d}{d x} e^{2 x}}{2} \]
   constant-multipleFactor out the constant.✓ Proved
5. \[ = - x \frac{d}{d x} e^{2 x} - e^{2 x} \frac{d}{d x} x + \frac{\frac{d}{d x} e^{2 x}}{2} \]
   productApply the product rule to the second term.✓ Proved
6. \[ = - 2 x e^{2 x} \]
   chain simplify algebra simplifyApply the chain rule to the exponential terms. Simplify the coefficients. Distribute the negative sign. Combine like terms.✓ Proved