Derivative of \( \displaystyle - x e^{x} + e^{x} \)

Problem 2.1935 · hard

1. \[ \frac{d}{d x} \left(- x e^{x} + e^{x}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- x e^{x}\right) + \frac{d}{d x} e^{x} \]
   sumApply the sum rule.✓ Proved
3. \[ = e^{x} + \frac{d}{d x} \left(- x e^{x}\right) \]
   derivativeDifferentiate the second term.✓ Proved
4. \[ = e^{x} - \frac{d}{d x} x e^{x} \]
   constantFactor out the negative sign.✓ Proved
5. \[ = - x \frac{d}{d x} e^{x} - e^{x} \frac{d}{d x} x + e^{x} \]
   productApply the product rule to the first term.✓ Proved
6. \[ = - x e^{x} \]
   derivative algebra simplifyDifferentiate the components of the product rule. Distribute the negative sign. Combine like terms.✓ Proved