Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(6 x + 2 \right)}}{12} \)

Problem 2.140 · hard

1. \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(6 x + 2 \right)}}{12}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(6 x + 2 \right)}}{12} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(6 x + 2 \right)}}{12} \]
   constantFactor out the constant 1/12.✓ Proved
4. \[ = - \frac{\cos{\left(6 x + 2 \right)} \frac{d}{d x} \left(6 x + 2\right)}{12} + \frac{d}{d x} \frac{x}{2} \]
   chainApply the chain rule to the sine term.✓ Proved
5. \[ = - \frac{\left(\frac{d}{d x} 2 + \frac{d}{d x} 6 x\right) \cos{\left(6 x + 2 \right)}}{12} + \frac{d}{d x} \frac{x}{2} \]
   sumApply the sum rule inside the derivative.✓ Proved
6. \[ = - \frac{\cos{\left(6 x + 2 \right)} \frac{d}{d x} x}{2} + \frac{d}{d x} \frac{x}{2} \]
   constantDifferentiate the constant 2.✓ Proved
7. \[ = - \frac{\cos{\left(6 x + 2 \right)}}{2} + \frac{d}{d x} \frac{x}{2} \]
   derivative algebraDifferentiate x. Multiply the constants.✓ Proved
8. \[ = \frac{1}{2} - \frac{\cos{\left(6 x + 2 \right)}}{2} \]
   simplifySimplify the final expression.✓ Proved