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

Problem 2.1435 · hard

1. \[ \frac{d}{d x} \frac{\left(3 \sin{\left(4 x \right)} - 3 \cos{\left(4 x \right)}\right) e^{4 x}}{8} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{3 \frac{d}{d x} \left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) e^{4 x}}{8} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = \frac{3 \left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) \frac{d}{d x} e^{4 x}}{8} + \frac{3 e^{4 x} \frac{d}{d x} \left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right)}{8} \]
   productApply the product rule.✓ Proved
4. \[ = \frac{3 \left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) e^{4 x}}{2} + \frac{3 \left(4 \sin{\left(4 x \right)} + 4 \cos{\left(4 x \right)}\right) e^{4 x}}{8} \]
   derivative algebraDifferentiate the trigonometric and exponential terms. Simplify the signs inside the first term.✓ Proved
5. \[ = 3 e^{4 x} \sin{\left(4 x \right)} \]
   algebra simplify algebra simplifyFactor out 4*exp(4*x) from both terms. Combine like terms inside the parentheses. Multiply the constants. Final simplified expression.✓ Proved