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

Problem 2.179 · hard

1. \[ \frac{d}{d x} \left(2 x - 1\right) e^{2 x} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(2 x e^{2 x} - e^{2 x}\right) \]
   algebraDistribute the exponential term.✓ Proved
3. \[ = \frac{d}{d x} 2 x e^{2 x} - \frac{d}{d x} e^{2 x} \]
   sumApply the sum rule for derivatives.✓ Proved
4. \[ = 2 \frac{d}{d x} x e^{2 x} - \frac{d}{d x} e^{2 x} \]
   constant-multiplePull out the constant factor 2.✓ Proved
5. \[ = 2 x \frac{d}{d x} e^{2 x} + 2 e^{2 x} \frac{d}{d x} x - \frac{d}{d x} e^{2 x} \]
   productApply the product rule to the first term.✓ Proved
6. \[ = 2 x \frac{d}{d x} e^{2 x} + 2 e^{2 x} - \frac{d}{d x} e^{2 x} \]
   derivativeDifferentiate x.✓ Proved
7. \[ = 4 x e^{2 x} \]
   chain algebra simplifyApply the chain rule to exp(2*x). Distribute the 2 into the parentheses. Combine like terms.✓ Proved