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

Problem 2.1490 · hard

1. \[ \frac{d}{d x} \left(x e^{x} - e^{x}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} x e^{x} - \frac{d}{d x} e^{x} \]
   sumApply the sum rule.✓ Proved
3. \[ = - e^{x} + \frac{d}{d x} x e^{x} \]
   exponentialDifferentiate the exponential term.✓ Proved
4. \[ = x \frac{d}{d x} e^{x} + e^{x} \frac{d}{d x} x - e^{x} \]
   productApply the product rule to the first term.✓ Proved
5. \[ = x e^{x} \]
   derivative algebra simplifyDifferentiate the individual components. Simplify the expression. Combine like terms.✓ Proved