Derivative of \( \displaystyle \frac{\left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{2} \)

Problem 2.319 · hard

1. \[ \frac{d}{d x} \frac{\left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{2} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) \frac{d}{d x} \frac{e^{2 x - 1}}{2} + \frac{e^{2 x - 1} \frac{d}{d x} \left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right)}{2} \]
   productApply the product rule.✓ Proved
3. \[ = \frac{\left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1} \frac{d}{d x} \left(2 x - 1\right)}{2} + \frac{e^{2 x - 1} \frac{d}{d x} \left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right)}{2} \]
   chainApply the chain rule to the exponential part.✓ Proved
4. \[ = \left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1} + \frac{e^{2 x - 1} \frac{d}{d x} \left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right)}{2} \]
   derivative algebraDifferentiate the inner function 2*x - 1. Simplify the constant factor 2/2.✓ Proved
5. \[ = \left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1} + \frac{\left(\frac{d}{d x} \sin{\left(2 x - 1 \right)} - \frac{d}{d x} \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{2} \]
   sumDistribute the derivative over the subtraction.✓ Proved
6. \[ = \frac{\left(\sin{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right) + \cos{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right)\right) e^{2 x - 1}}{2} + \left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1} \]
   trigDifferentiate the sine and cosine terms using the chain rule.✓ Proved
7. \[ = \left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1} + \frac{\left(2 \sin{\left(2 x - 1 \right)} + 2 \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{2} \]
   derivative algebraDifferentiate the inner function 2*x - 1. Factor out the 2.✓ Proved
8. \[ = \left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1} + \left(\sin{\left(2 x - 1 \right)} + \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1} \]
   algebraCancel the 2 in the numerator and denominator.✓ Proved
9. \[ = 2 e^{2 x - 1} \sin{\left(2 x - 1 \right)} \]
   algebra simplify algebraFactor out the common exponential term. Combine like terms inside the parentheses. Rearrange the terms for the final answer.✓ Proved