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

Problem 2.33 · hard

1. \[ \frac{d}{d x} \frac{\left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) e^{4 x}}{8} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) e^{4 x}}{8} \]
   constant-multiplePull out the constant factor 1/8.✓ Proved
3. \[ = \frac{\left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) \frac{d}{d x} e^{4 x}}{8} + \frac{e^{4 x} \frac{d}{d x} \left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right)}{8} \]
   productApply the product rule to the numerator.✓ Proved
4. \[ = \frac{\left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) \frac{d}{d x} e^{4 x}}{8} + \frac{\left(\frac{d}{d x} \sin{\left(4 x \right)} - \frac{d}{d x} \cos{\left(4 x \right)}\right) e^{4 x}}{8} \]
   sumApply the sum rule to the first term.✓ Proved
5. \[ = \frac{\left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) e^{4 x}}{2} + \frac{\left(4 \sin{\left(4 x \right)} + 4 \cos{\left(4 x \right)}\right) e^{4 x}}{8} \]
   trig algebraDifferentiate the sine and cosine terms using the chain rule. Simplify the signs inside the parentheses.✓ Proved
6. \[ = e^{4 x} \sin{\left(4 x \right)} \]
   algebra algebra simplify simplifyActually, let's just distribute and simplify. Distribute the terms. Combine like terms. Simplify the final expression.✓ Proved