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

Problem 2.625 · hard

1. \[ \frac{d}{d x} e^{3 - x} \sin{\left(x - 3 \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = e^{3 - x} \frac{d}{d x} \sin{\left(x - 3 \right)} + \sin{\left(x - 3 \right)} \frac{d}{d x} e^{3 - x} \]
   productApply the product rule.✓ Proved
3. \[ = - e^{3 - x} \sin{\left(x - 3 \right)} + e^{3 - x} \frac{d}{d x} \sin{\left(x - 3 \right)} \]
   chainDifferentiate the first part using the chain rule.✓ Proved
4. \[ = - e^{3 - x} \sin{\left(x - 3 \right)} + e^{3 - x} \cos{\left(x - 3 \right)} \]
   chain algebraDifferentiate the second part using the chain rule. Simplify the expression by multiplying constants.✓ Proved
5. \[ = \left(- \sin{\left(x - 3 \right)} + \cos{\left(x - 3 \right)}\right) e^{3 - x} \]
   simplifyFactor out the common term.✓ Proved