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

Problem 2.869 · medium

1. \[ \frac{d}{d x} \left(1 - x\right) e^{x} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \left(1 - x\right) \frac{d}{d x} e^{x} + e^{x} \frac{d}{d x} \left(1 - x\right) \]
   productApply the product rule.✓ Proved
3. \[ = \left(1 - x\right) \frac{d}{d x} e^{x} + e^{x} \frac{d}{d x} 1 - e^{x} \frac{d}{d x} x \]
   algebraDistribute the derivative over the subtraction.✓ Proved
4. \[ = \left(1 - x\right) \frac{d}{d x} e^{x} - e^{x} \]
   derivative algebraDifferentiate the terms inside the parentheses. Simplify the expression.✓ Proved
5. \[ = \left(1 - x\right) e^{x} - e^{x} \]
   derivativeDifferentiate the exponential term.✓ Proved
6. \[ = - x e^{x} \]
   algebra simplifyDistribute the exponential term. Combine like terms.✓ Proved