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

Problem 2.369 · hard

1. \[ \frac{d}{d x} e^{- x} \sin{\left(x \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \sin{\left(x \right)} \frac{d}{d x} e^{- x} + e^{- x} \frac{d}{d x} \sin{\left(x \right)} \]
   productApply the product rule.✓ Proved
3. \[ = - e^{- x} \sin{\left(x \right)} + e^{- x} \frac{d}{d x} \sin{\left(x \right)} \]
   chainDifferentiate the exponential part using the chain rule.✓ Proved
4. \[ = - e^{- x} \sin{\left(x \right)} + e^{- x} \cos{\left(x \right)} \]
   derivativeDifferentiate the sine part.✓ Proved
5. \[ = \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{- x} \]
   algebraFactor out the common term exp(-x).✓ Proved