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

Problem 2.498 · medium

1. \[ \frac{d}{d x} \left(x + \frac{1}{2}\right) e^{2 x + 2} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} x e^{2 x + 2} + \frac{d}{d x} \frac{e^{2 x + 2}}{2} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} x e^{2 x + 2} + \frac{\frac{d}{d x} e^{2 x + 2}}{2} \]
   constant-multiplePull out the constant 1/2.✓ Proved
4. \[ = x \frac{d}{d x} e^{2 x + 2} + e^{2 x + 2} \frac{d}{d x} x + \frac{\frac{d}{d x} e^{2 x + 2}}{2} \]
   productApply the product rule to the first term.✓ Proved
5. \[ = x \frac{d}{d x} e^{2 x + 2} + e^{2 x + 2} + \frac{\frac{d}{d x} e^{2 x + 2}}{2} \]
   derivativeDifferentiate x.✓ Proved
6. \[ = x e^{2 x + 2} \frac{d}{d x} \left(2 x + 2\right) + \frac{e^{2 x + 2} \frac{d}{d x} \left(2 x + 2\right)}{2} + e^{2 x + 2} \]
   chainApply the chain rule to the exponential terms.✓ Proved
7. \[ = 2 x e^{2 x + 2} + 2 e^{2 x + 2} \]
   derivative algebra algebraDifferentiate the inner function 2*x + 2. Simplify the multiplication. Combine like terms.✓ Proved
8. \[ = \left(2 x + 2\right) e^{2 x + 2} \]
   simplifyFactor out the common term.✓ Proved