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

Problem 2.1489 · hard

1. \[ \frac{d}{d x} \left(\frac{e^{4 x} \sin{\left(4 x \right)}}{8} - \frac{e^{4 x} \cos{\left(4 x \right)}}{8}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \frac{e^{4 x} \sin{\left(4 x \right)}}{8} - \frac{d}{d x} \frac{e^{4 x} \cos{\left(4 x \right)}}{8} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{\frac{d}{d x} e^{4 x} \sin{\left(4 x \right)}}{8} - \frac{\frac{d}{d x} e^{4 x} \cos{\left(4 x \right)}}{8} \]
   constant-multiple algebraFactor out the constant 1/8. Factor out 1/8 from the expression.✓ Proved
4. \[ = \frac{e^{4 x} \frac{d}{d x} \sin{\left(4 x \right)}}{8} - \frac{e^{4 x} \frac{d}{d x} \cos{\left(4 x \right)}}{8} + \frac{\sin{\left(4 x \right)} \frac{d}{d x} e^{4 x}}{8} - \frac{\cos{\left(4 x \right)} \frac{d}{d x} e^{4 x}}{8} \]
   productApply the product rule to both terms.✓ Proved
5. \[ = \frac{e^{4 x} \sin{\left(4 x \right)}}{2} - \frac{e^{4 x} \cos{\left(4 x \right)}}{2} + \frac{e^{4 x} \frac{d}{d x} \sin{\left(4 x \right)}}{8} - \frac{e^{4 x} \frac{d}{d x} \cos{\left(4 x \right)}}{8} \]
   derivativeDifferentiate the exponential terms.✓ Proved
6. \[ = e^{4 x} \sin{\left(4 x \right)} \]
   trig algebra algebra simplify simplifyDifferentiate the trigonometric terms. Simplify the expression inside the parentheses. Distribute the negative sign. Combine like terms. Simplify the final expression.✓ Proved