Derivative of \( \displaystyle \frac{\left(3 \sin{\left(4 x + 1 \right)} - 3 \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} \)

Problem 2.109 · hard

1. \[ \frac{d}{d x} \frac{\left(3 \sin{\left(4 x + 1 \right)} - 3 \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} \]
   constantStart with the derivative of the function.✓ Proved
2. \[ = \frac{3 \frac{d}{d x} \left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = \frac{3 \left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) \frac{d}{d x} e^{4 x + 1}}{8} + \frac{3 e^{4 x + 1} \frac{d}{d x} \left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right)}{8} \]
   productApply the product rule.✓ Proved
4. \[ = \frac{3 \left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) \frac{d}{d x} e^{4 x + 1}}{8} + \frac{3 \left(\frac{d}{d x} \sin{\left(4 x + 1 \right)} - \frac{d}{d x} \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} \]
   sumDistribute the derivative over the difference.✓ Proved
5. \[ = \frac{3 \left(4 \sin{\left(4 x + 1 \right)} - 4 \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} + \frac{3 \left(4 \sin{\left(4 x + 1 \right)} + 4 \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} \]
   chain algebraApply the chain rule to each term. Simplify the signs.✓ Proved
6. \[ = 3 e^{4 x + 1} \sin{\left(4 x + 1 \right)} \]
   algebra simplify simplifyDistribute the exponential term. Combine like terms. Final simplification.✓ Proved