Derivative of \( \displaystyle \frac{\left(5 \sin{\left(4 x - 1 \right)} - 5 \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \)

Problem 2.594 · hard

1. \[ \frac{d}{d x} \frac{\left(5 \sin{\left(4 x - 1 \right)} - 5 \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \]
   constant-multiplePull out the constant factor.✓ Proved
2. \[ = \frac{5 \frac{d}{d x} \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \]
   productApply the product rule.✓ Proved
3. \[ = \frac{5 \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) \frac{d}{d x} e^{4 x - 1}}{8} + \frac{5 e^{4 x - 1} \frac{d}{d x} \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right)}{8} \]
   sumDifferentiate the terms inside the parentheses.✓ Proved
4. \[ = \frac{5 \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) \frac{d}{d x} e^{4 x - 1}}{8} + \frac{5 \left(\frac{d}{d x} \sin{\left(4 x - 1 \right)} - \frac{d}{d x} \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \]
   chainApply the chain rule to the sine and cosine terms.✓ Proved
5. \[ = \frac{5 \left(4 \sin{\left(4 x - 1 \right)} - 4 \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} + \frac{5 \left(4 \sin{\left(4 x - 1 \right)} + 4 \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \]
   trig algebraEvaluate the derivatives of sine and cosine. Simplify the signs.✓ Proved
6. \[ = 5 e^{4 x - 1} \sin{\left(4 x - 1 \right)} \]
   algebra simplify simplifyDistribute the exponential term. Combine like terms. Final simplification.✓ Proved