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

Problem 2.1030 · hard

1. \[ \frac{d}{d x} \frac{\left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) \frac{d}{d x} \frac{e^{4 x + 1}}{8} + \frac{e^{4 x + 1} \frac{d}{d x} \left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right)}{8} \]
   productApply the product rule.✓ Proved
3. \[ = \frac{\left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) \frac{d}{d x} e^{4 x + 1}}{8} + \frac{e^{4 x + 1} \frac{d}{d x} \left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right)}{8} \]
   constantPull out the constant factor 1/8.✓ Proved
4. \[ = \frac{\left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) \frac{d}{d x} e^{4 x + 1}}{8} + \frac{\left(\frac{d}{d x} \sin{\left(4 x + 1 \right)} - \frac{d}{d x} \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} \]
   sumApply the sum rule to the first term.✓ Proved
5. \[ = \frac{\left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{2} + \frac{\left(4 \sin{\left(4 x + 1 \right)} + 4 \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} \]
   chain algebraApply the chain rule to the trigonometric and exponential terms. Simplify the signs.✓ Proved
6. \[ = \frac{\left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{2} + \frac{\left(\sin{\left(4 x + 1 \right)} + \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{2} \]
   simplifySimplify the coefficients.✓ Proved
7. \[ = e^{4 x + 1} \sin{\left(4 x + 1 \right)} \]
   algebra simplify simplifyFactor out the common term exp(4*x + 1) / 2. Combine like terms inside the parentheses. Final simplification.✓ Proved