Derivative of \( \displaystyle - \frac{x}{2} + \frac{\sin{\left(6 x + 4 \right)}}{12} \)

Problem 2.696 · hard

1. \[ \frac{d}{d x} \left(- \frac{x}{2} + \frac{\sin{\left(6 x + 4 \right)}}{12}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \frac{\sin{\left(6 x + 4 \right)}}{12} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{\frac{d}{d x} \sin{\left(6 x + 4 \right)}}{12} \]
   constantPull out the constant factor.✓ Proved
4. \[ = \frac{\cos{\left(6 x + 4 \right)} \frac{d}{d x} \left(6 x + 4\right)}{12} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]
   chainApply the chain rule to the sine function.✓ Proved
5. \[ = \frac{\cos{\left(6 x + 4 \right)}}{2} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]
   derivative algebraDifferentiate the inner function 6*x + 4. Simplify the constant multiplication.✓ Proved
6. \[ = \frac{\cos{\left(6 x + 4 \right)}}{2} - \frac{1}{2} \]
   derivativeEvaluate the derivative of the first term.✓ Proved