Derivative of \( \displaystyle \left(2 - x\right) e^{x - 1} \)

Problem 2.1066 · medium

1. \[ \frac{d}{d x} \left(2 - x\right) e^{x - 1} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- x e^{x - 1} + 2 e^{x - 1}\right) \]
   algebraDistribute the term (2 - x).✓ Proved
3. \[ = - \frac{d}{d x} x e^{x - 1} + \frac{d}{d x} 2 e^{x - 1} \]
   sumApply the difference rule.✓ Proved
4. \[ = - \frac{d}{d x} x e^{x - 1} + 2 \frac{d}{d x} e^{x - 1} \]
   constant-multiplePull out the constant factor 2.✓ Proved
5. \[ = - x \frac{d}{d x} e^{x - 1} - e^{x - 1} \frac{d}{d x} x + 2 \frac{d}{d x} e^{x - 1} \]
   productApply the product rule to the second term.✓ Proved
6. \[ = - x \frac{d}{d x} e^{x - 1} - e^{x - 1} + 2 \frac{d}{d x} e^{x - 1} \]
   derivativeDifferentiate x.✓ Proved
7. \[ = - x e^{x - 1} \frac{d}{d x} \left(x - 1\right) + 2 e^{x - 1} \frac{d}{d x} \left(x - 1\right) - e^{x - 1} \]
   chainApply the chain rule to exp(x - 1).✓ Proved
8. \[ = - x e^{x - 1} + e^{x - 1} \]
   derivative algebra algebra simplifyDifferentiate x - 1. Simplify the expression. Distribute the negative sign. Combine like terms.✓ Proved
9. \[ = \left(1 - x\right) e^{x - 1} \]
   simplifyFactor out the common term exp(x - 1).✓ Proved