Derivative of \( \displaystyle \frac{\left(5 \sin{\left(2 x + 2 \right)} - 5 \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \)

Problem 2.1199 · hard

1. \[ \frac{d}{d x} \frac{\left(5 \sin{\left(2 x + 2 \right)} - 5 \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{5 \frac{d}{d x} \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = \frac{5 \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) \frac{d}{d x} e^{2 x + 2}}{4} + \frac{5 e^{2 x + 2} \frac{d}{d x} \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right)}{4} \]
   productApply the product rule.✓ Proved
4. \[ = \frac{5 \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) \frac{d}{d x} e^{2 x + 2}}{4} + \frac{5 \left(\frac{d}{d x} \sin{\left(2 x + 2 \right)} - \frac{d}{d x} \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \]
   sumDistribute the derivative over the subtraction.✓ Proved
5. \[ = \frac{5 \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{2} + \frac{5 \left(2 \sin{\left(2 x + 2 \right)} + 2 \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \]
   chainApply the chain rule to the trigonometric and exponential terms.✓ Proved
6. \[ = \frac{5 \left(2 \sin{\left(2 x + 2 \right)} - 2 \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} + \frac{5 \left(2 \sin{\left(2 x + 2 \right)} + 2 \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \]
   algebraSimplify the signs inside the first parenthesis.✓ Proved
7. \[ = 5 e^{2 x + 2} \sin{\left(2 x + 2 \right)} \]
   algebra algebra simplifyDistribute the exponential term. Combine like terms. Final simplification.✓ Proved