Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(2 x - 6 \right)}}{4} \)

Problem 2.904 · hard

1. \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(2 x - 6 \right)}}{4}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(2 x - 6 \right)}}{4} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(2 x - 6 \right)}}{4} \]
   constant-multipleFactor out the constant 1/4.✓ Proved
4. \[ = - \frac{\cos{\left(2 x - 6 \right)} \frac{d}{d x} \left(2 x - 6\right)}{4} + \frac{d}{d x} \frac{x}{2} \]
   chainApply the chain rule to the sine term.✓ Proved
5. \[ = - \frac{\cos{\left(2 x - 6 \right)} \frac{d}{d x} 6}{4} - \frac{\cos{\left(2 x - 6 \right)} \frac{d}{d x} 2 x}{4} + \frac{d}{d x} \frac{x}{2} \]
   sumApply the sum rule to the inner function.✓ Proved
6. \[ = - \frac{\cos{\left(2 x - 6 \right)}}{2} + \frac{d}{d x} \frac{x}{2} \]
   constant algebraThe derivative of the constant 6 is 0. Simplify the expression.✓ Proved
7. \[ = \frac{1}{2} - \frac{\cos{\left(2 x - 6 \right)}}{2} \]
   derivative simplifyDifferentiate the first term. Simplify the final expression.✓ Proved