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

Problem 2.1258 · hard

1. \[ \frac{d}{d x} \left(- \frac{x}{2} + \frac{\sin{\left(6 x \right)}}{12}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \frac{\sin{\left(6 x \right)}}{12} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{\frac{d}{d x} \sin{\left(6 x \right)}}{12} \]
   constant-multipleFactor out the constant 1/12.✓ Proved
4. \[ = \frac{\cos{\left(6 x \right)} \frac{d}{d x} 6 x}{12} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]
   chainApply the chain rule to sin(6*x).✓ Proved
5. \[ = \frac{\cos{\left(6 x \right)}}{2} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]
   derivative algebra simplifyDifferentiate 6*x. Multiply the constants. Simplify the fraction.✓ Proved
6. \[ = \frac{\cos{\left(6 x \right)}}{2} - \frac{1}{2} \]
   derivative simplifyDifferentiate the remaining term. Final simplified form.✓ Proved