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

Problem 2.421 · hard

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