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

Problem 2.1484 · hard

1. \[ \frac{d}{d x} \ln{\left(x e^{- x} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- x + \ln{\left(x \right)}\right) \]
   algebra simplifyUse the logarithm product rule to split the terms. Simplify the logarithm of the exponential term.✓ Proved
3. \[ = - \frac{d}{d x} x + \frac{d}{d x} \ln{\left(x \right)} \]
   sumApply the linearity of the derivative.✓ Proved
4. \[ = \frac{d}{d x} \ln{\left(x \right)} - 1 \]
   derivativeDifferentiate the term x.✓ Proved
5. \[ = -1 + \frac{1}{x} \]
   derivativeDifferentiate the logarithm term.✓ Proved
6. \[ = \frac{1 - x}{x} \]
   simplifyCombine the terms into a single fraction.✓ Proved