Derivative of \( \displaystyle \frac{\left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{2} \)

Problem 2.656 · hard

1. \[ \frac{d}{d x} \frac{\left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{2} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{2} \]
   constant-multiplePull out the constant factor 1/2.✓ Proved
3. \[ = \frac{\left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) \frac{d}{d x} e^{2 x - 3}}{2} + \frac{e^{2 x - 3} \frac{d}{d x} \left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right)}{2} \]
   productApply the product rule.✓ Proved
4. \[ = \frac{\left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) \frac{d}{d x} e^{2 x - 3}}{2} + \frac{\left(\frac{d}{d x} \sin{\left(2 x - 3 \right)} - \frac{d}{d x} \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{2} \]
   sumApply the sum rule to the first term.✓ Proved
5. \[ = \frac{\left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) \frac{d}{d x} e^{2 x - 3}}{2} + \frac{\left(2 \sin{\left(2 x - 3 \right)} + 2 \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{2} \]
   trig algebraDifferentiate the sine and cosine terms using the chain rule. Distribute the 2 into the parentheses.✓ Proved
6. \[ = \left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3} + \frac{\left(2 \sin{\left(2 x - 3 \right)} + 2 \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{2} \]
   exponentialDifferentiate the exponential term.✓ Proved
7. \[ = 2 e^{2 x - 3} \sin{\left(2 x - 3 \right)} \]
   algebra algebra simplify simplifyFactor out the common term exp(2*x - 3). Distribute the 2. Combine like terms. Simplify the final expression.✓ Proved