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

Problem 2.884 · 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. \[ = \left(- x - \frac{1}{2}\right) \frac{d}{d x} e^{2 x + 2} + e^{2 x + 2} \frac{d}{d x} \left(- x - \frac{1}{2}\right) \]
   productApply the product rule.✓ Proved
3. \[ = \left(- x - \frac{1}{2}\right) \frac{d}{d x} e^{2 x + 2} + e^{2 x + 2} \frac{d}{d x} \left(- \frac{1}{2}\right) + e^{2 x + 2} \frac{d}{d x} \left(- x\right) \]
   sumApply the sum rule to the first term.✓ Proved
4. \[ = \left(- x - \frac{1}{2}\right) \frac{d}{d x} e^{2 x + 2} - e^{2 x + 2} \]
   derivativeDifferentiate the individual terms.✓ Proved
5. \[ = - \left(x + \frac{1}{2}\right) \frac{d}{d x} e^{2 x + 2} - e^{2 x + 2} \]
   algebraSimplify the expression.✓ Proved
6. \[ = - \left(x + \frac{1}{2}\right) e^{2 x + 2} \frac{d}{d x} \left(2 x + 2\right) - e^{2 x + 2} \]
   exponentialDifferentiate the exponential term using the chain rule.✓ Proved
7. \[ = - 2 \left(x + \frac{1}{2}\right) e^{2 x + 2} - e^{2 x + 2} \]
   derivativeDifferentiate the exponent.✓ Proved
8. \[ = - \left(2 x + 1\right) e^{2 x + 2} - e^{2 x + 2} \]
   algebraDistribute the constant and simplify.✓ Proved
9. \[ = - 2 x e^{2 x + 2} - 2 e^{2 x + 2} \]
   algebra algebraDistribute the term. Combine like terms.✓ Proved
10. \[ = \left(- 2 x - 2\right) e^{2 x + 2} \]
    simplifyFactor out the common term.✓ Proved