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

Problem 2.948 · medium

1. \[ \frac{d}{d x} \left(4 - x\right) e^{x - 3} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \left(4 - x\right) \frac{d}{d x} e^{x - 3} + e^{x - 3} \frac{d}{d x} \left(4 - x\right) \]
   productApply the product rule.✓ Proved
3. \[ = \left(4 - x\right) \frac{d}{d x} e^{x - 3} - e^{x - 3} \frac{d}{d x} x + \frac{d}{d x} 4 \]
   sumDifferentiate the terms in the first part of the product rule.✓ Proved
4. \[ = \left(4 - x\right) \frac{d}{d x} e^{x - 3} - e^{x - 3} \]
   derivative simplifyEvaluate the derivatives of the individual terms. Simplify the expression.✓ Proved
5. \[ = \left(4 - x\right) e^{x - 3} \frac{d}{d x} \left(x - 3\right) - e^{x - 3} \]
   chainApply the chain rule to the exponential term.✓ Proved
6. \[ = \left(4 - x\right) e^{x - 3} - e^{x - 3} \]
   derivative simplifyEvaluate the derivative of the inner function. Simplify the expression.✓ Proved
7. \[ = \left(3 - x\right) e^{x - 3} \]
   algebra simplifyFactor out the common exponential term. Combine like terms.✓ Proved