Derivative of \( \displaystyle \frac{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x}}{2} \)

Problem 2.1566 · hard Beautiful

1. \[ \frac{d}{d x} \frac{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x}}{2} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x}}{2} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = \frac{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) \frac{d}{d x} e^{x}}{2} + \frac{e^{x} \frac{d}{d x} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)}{2} \]
   productApply the product rule.✓ Proved
4. \[ = \frac{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) \frac{d}{d x} e^{x}}{2} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}}{2} \]
   derivative algebraDifferentiate the first part of the product. Simplify the signs.✓ Proved
5. \[ = \frac{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x}}{2} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}}{2} \]
   derivativeDifferentiate the second part of the product.✓ Proved
6. \[ = e^{x} \sin{\left(x \right)} \]
   algebra simplify simplifyFactor out exp(x). Combine like terms. Final simplification.✓ Proved