Derivative of \( \displaystyle \left(x - 4\right) e^{x - 3} \)

Problem 2.1190 · medium

1. \[ \frac{d}{d x} \left(x - 4\right) e^{x - 3} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(x e^{x - 3} - 4 e^{x - 3}\right) \]
   algebraDistribute the term (x - 4).✓ Proved
3. \[ = \frac{d}{d x} x e^{x - 3} - \frac{d}{d x} 4 e^{x - 3} \]
   sumApply the sum rule for derivatives.✓ Proved
4. \[ = \frac{d}{d x} x e^{x - 3} - 4 \frac{d}{d x} e^{x - 3} \]
   constantPull out the constant factor 4.✓ Proved
5. \[ = x \frac{d}{d x} e^{x - 3} + e^{x - 3} \frac{d}{d x} x - 4 \frac{d}{d x} e^{x - 3} \]
   productApply the product rule to the first term.✓ Proved
6. \[ = x \frac{d}{d x} e^{x - 3} + e^{x - 3} - 4 \frac{d}{d x} e^{x - 3} \]
   derivativeDifferentiate x.✓ Proved
7. \[ = x e^{x - 3} - 3 e^{x - 3} \]
   exponentialDifferentiate the exponential term.✓ Proved
8. \[ = \left(x - 4\right) e^{x - 3} + e^{x - 3} \]
   algebraCombine the terms involving exp(x - 3).✓ Proved
9. \[ = \left(x - 3\right) e^{x - 3} \]
   algebra simplifyFactor out exp(x - 3). Simplify the expression.✓ Proved