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

Problem 2.1178 · hard

1. \[ \frac{d}{d x} \frac{\left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) e^{4 x}}{4} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) e^{4 x}}{4} \]
   constant-multiplePull out the constant factor 1/4.✓ Proved
3. \[ = \frac{\left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) \frac{d}{d x} e^{4 x}}{4} + \frac{e^{4 x} \frac{d}{d x} \left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right)}{4} \]
   productApply the product rule to the numerator.✓ Proved
4. \[ = \frac{\left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) \frac{d}{d x} e^{4 x}}{4} + \frac{\left(\frac{d}{d x} \sin{\left(4 x \right)} - \frac{d}{d x} \cos{\left(4 x \right)}\right) e^{4 x}}{4} \]
   sumApply the sum rule to the first term.✓ Proved
5. \[ = \frac{\left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) \frac{d}{d x} e^{4 x}}{4} + \frac{\left(4 \sin{\left(4 x \right)} + 4 \cos{\left(4 x \right)}\right) e^{4 x}}{4} \]
   trig algebraDifferentiate the sine and cosine terms. Simplify the signs.✓ Proved
6. \[ = \left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) e^{4 x} + \frac{\left(4 \sin{\left(4 x \right)} + 4 \cos{\left(4 x \right)}\right) e^{4 x}}{4} \]
   exponentialDifferentiate the exponential term.✓ Proved
7. \[ = 2 e^{4 x} \sin{\left(4 x \right)} \]
   algebra simplify simplifyFactor out 4*exp(4*x) from the terms. Combine like terms inside the parentheses. Simplify the final expression.✓ Proved