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

Problem 2.182 · hard

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