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

Problem 2.1939 · hard

1. \[ \frac{d}{d x} \left(- \frac{e^{4 x} \sin{\left(4 x \right)}}{8} + \frac{e^{4 x} \cos{\left(4 x \right)}}{8}\right) \]
   sumStart with the derivative of the sum.✓ Proved
2. \[ = \frac{d}{d x} \left(- \frac{e^{4 x} \sin{\left(4 x \right)}}{8}\right) + \frac{d}{d x} \frac{e^{4 x} \cos{\left(4 x \right)}}{8} \]
   constant-multiplePull out the constant factors.✓ Proved
3. \[ = - \frac{\frac{d}{d x} e^{4 x} \sin{\left(4 x \right)}}{8} + \frac{\frac{d}{d x} e^{4 x} \cos{\left(4 x \right)}}{8} \]
   productApply the product rule to each term.✓ Proved
4. \[ = - \frac{e^{4 x} \frac{d}{d x} \sin{\left(4 x \right)}}{8} + \frac{e^{4 x} \frac{d}{d x} \cos{\left(4 x \right)}}{8} - \frac{\sin{\left(4 x \right)} \frac{d}{d x} e^{4 x}}{8} + \frac{\cos{\left(4 x \right)} \frac{d}{d x} e^{4 x}}{8} \]
   exponentialDifferentiate the exponential functions.✓ Proved
5. \[ = - \frac{e^{4 x} \sin{\left(4 x \right)}}{2} + \frac{e^{4 x} \cos{\left(4 x \right)}}{2} - \frac{e^{4 x} \frac{d}{d x} \sin{\left(4 x \right)}}{8} + \frac{e^{4 x} \frac{d}{d x} \cos{\left(4 x \right)}}{8} \]
   trigDifferentiate the trigonometric functions.✓ Proved
6. \[ = - e^{4 x} \sin{\left(4 x \right)} \]
   algebra simplify algebra simplify simplifyDistribute the constants inside the parentheses. Simplify the coefficients. Combine like terms. Cancel the cosine terms. Combine the remaining sine terms.✓ Proved