Derivative of \( \displaystyle \frac{3 e^{4 x} \sin{\left(4 x \right)}}{8} - \frac{3 e^{4 x} \cos{\left(4 x \right)}}{8} \)

Problem 2.1856 · hard

1. \[ \frac{d}{d x} \left(\frac{3 e^{4 x} \sin{\left(4 x \right)}}{8} - \frac{3 e^{4 x} \cos{\left(4 x \right)}}{8}\right) \]
   derivative constant-multipleStart with the derivative of the function. Factor out the constant 3/8 from both terms.✓ Proved
2. \[ = \frac{3 \frac{d}{d x} e^{4 x} \sin{\left(4 x \right)}}{8} - \frac{3 \frac{d}{d x} e^{4 x} \cos{\left(4 x \right)}}{8} \]
   sumApply the sum rule for derivatives.✓ Proved
3. \[ = - \frac{3 e^{4 x} \frac{d}{d x} \cos{\left(4 x \right)}}{8} - \frac{3 \cos{\left(4 x \right)} \frac{d}{d x} e^{4 x}}{8} + \frac{3 \frac{d}{d x} e^{4 x} \sin{\left(4 x \right)}}{8} \]
   productApply the product rule to the second term.✓ Proved
4. \[ = \frac{3 e^{4 x} \frac{d}{d x} \sin{\left(4 x \right)}}{8} - \frac{3 e^{4 x} \frac{d}{d x} \cos{\left(4 x \right)}}{8} + \frac{3 \sin{\left(4 x \right)} \frac{d}{d x} e^{4 x}}{8} - \frac{3 \cos{\left(4 x \right)} \frac{d}{d x} e^{4 x}}{8} \]
   productApply the product rule to the first term.✓ Proved
5. \[ = 3 e^{4 x} \sin{\left(4 x \right)} \]
   chain algebra algebra simplify simplifyDifferentiate the exponential and trigonometric functions using the chain rule. Distribute the negative sign. Simplify the double negative. Combine like terms. Final simplification.✓ Proved