Derivative of \( \displaystyle \frac{\left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \)

Problem 2.897 · hard

1. \[ \frac{d}{d x} \frac{\left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) \frac{d}{d x} \frac{e^{4 x - 1}}{8} + \frac{e^{4 x - 1} \frac{d}{d x} \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right)}{8} \]
   productApply the product rule.✓ Proved
3. \[ = \frac{\left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1} \frac{d}{d x} \left(4 x - 1\right)}{8} + \frac{e^{4 x - 1} \frac{d}{d x} \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right)}{8} \]
   chainApply the chain rule to the second term.✓ Proved
4. \[ = \frac{\left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{2} + \frac{e^{4 x - 1} \frac{d}{d x} \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right)}{8} \]
   derivativeDifferentiate the inner function 4*x - 1.✓ Proved
5. \[ = \frac{\left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{2} + \frac{\left(4 \sin{\left(4 x - 1 \right)} + 4 \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \]
   derivative algebraDifferentiate the sine and cosine terms. Simplify the first term's parentheses.✓ Proved
6. \[ = \frac{\left(4 \sin{\left(4 x - 1 \right)} - 4 \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} + \frac{\left(4 \sin{\left(4 x - 1 \right)} + 4 \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \]
   constant-multipleFactor out the 4 from both terms.✓ Proved
7. \[ = e^{4 x - 1} \sin{\left(4 x - 1 \right)} \]
   algebra algebra simplify simplifyFactor out the common term exp(4*x - 1)/8. Distribute the 4 in the second term. Combine like terms in the numerator. Simplify the fraction by canceling the 8.✓ Proved