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

Problem 2.2170 · hard

1. \[ \frac{d}{d x} \left(\frac{e^{4 x} \sin{\left(4 x \right)}}{4} - \frac{e^{4 x} \cos{\left(4 x \right)}}{4}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \frac{e^{4 x} \sin{\left(4 x \right)}}{4} - \frac{d}{d x} \frac{e^{4 x} \cos{\left(4 x \right)}}{4} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{\frac{d}{d x} e^{4 x} \sin{\left(4 x \right)}}{4} - \frac{\frac{d}{d x} e^{4 x} \cos{\left(4 x \right)}}{4} \]
   constant-multiple algebraFactor out the constant 1/4. Factor out 1/4 from both terms.✓ Proved
4. \[ = \frac{e^{4 x} \frac{d}{d x} \sin{\left(4 x \right)}}{4} - \frac{e^{4 x} \frac{d}{d x} \cos{\left(4 x \right)}}{4} + \frac{\sin{\left(4 x \right)} \frac{d}{d x} e^{4 x}}{4} - \frac{\cos{\left(4 x \right)} \frac{d}{d x} e^{4 x}}{4} \]
   productApply the product rule to both terms.✓ Proved
5. \[ = e^{4 x} \sin{\left(4 x \right)} - e^{4 x} \cos{\left(4 x \right)} + \frac{e^{4 x} \frac{d}{d x} \sin{\left(4 x \right)}}{4} - \frac{e^{4 x} \frac{d}{d x} \cos{\left(4 x \right)}}{4} \]
   exponentialDifferentiate the exponential term.✓ Proved
6. \[ = 2 e^{4 x} \sin{\left(4 x \right)} \]
   trig algebra algebra simplify simplifyDifferentiate the trigonometric terms. Distribute the terms inside the parentheses. Remove the inner parentheses. Combine like terms. Final simplification.✓ Proved