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

Problem 2.362 · hard

1. \[ \frac{d}{d x} x e^{- x} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = x \frac{d}{d x} e^{- x} + e^{- x} \frac{d}{d x} x \]
   productApply the product rule.✓ Proved
3. \[ = x \frac{d}{d x} e^{- x} + e^{- x} \]
   derivative simplifyDifferentiate the first part of the product. Simplify the first term.✓ Proved
4. \[ = x e^{- x} \frac{d}{d x} \left(- x\right) + e^{- x} \]
   chainApply the chain rule to the exponential term.✓ Proved
5. \[ = - x e^{- x} + e^{- x} \]
   derivative algebraDifferentiate the exponent. Simplify the expression.✓ Proved
6. \[ = \left(1 - x\right) e^{- x} \]
   simplifyFactor out the common term.✓ Proved