Derivative of \( \displaystyle \frac{3 x}{2} - \frac{3 \sin{\left(4 x - 2 \right)}}{8} \)

Problem 2.673 · hard

1. \[ \frac{d}{d x} \left(\frac{3 x}{2} - \frac{3 \sin{\left(4 x - 2 \right)}}{8}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \frac{3 x}{2} - \frac{d}{d x} \frac{3 \sin{\left(4 x - 2 \right)}}{8} \]
   sum constant-multipleApply the sum rule. Factor out the constants.✓ Proved
3. \[ = \frac{3 \frac{d}{d x} x}{2} - \frac{3 \frac{d}{d x} \sin{\left(4 x - 2 \right)}}{8} \]
   constant-multipleApply the constant multiple rule to both terms.✓ Proved
4. \[ = \frac{3}{2} - \frac{3 \frac{d}{d x} \sin{\left(4 x - 2 \right)}}{8} \]
   derivative algebraDifferentiate the first term. Simplify the first term.✓ Proved
5. \[ = - \frac{3 \cos{\left(4 x - 2 \right)} \frac{d}{d x} \left(4 x - 2\right)}{8} + \frac{3}{2} \]
   trigApply the derivative rule for sine.✓ Proved
6. \[ = - \frac{3 \cos{\left(4 x - 2 \right)} \frac{d}{d x} x}{2} + \frac{3}{2} \]
   chainApply the chain rule to the inner function.✓ Proved
7. \[ = \frac{3}{2} - \frac{3 \cos{\left(4 x - 2 \right)}}{2} \]
   derivative algebraDifferentiate the inner function. Simplify the expression.✓ Proved