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

Problem 2.1100 · hard

1. \[ \frac{d}{d x} \left(x - \frac{\sin{\left(6 x + 2 \right)}}{6}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} x - \frac{d}{d x} \frac{\sin{\left(6 x + 2 \right)}}{6} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} x - \frac{\frac{d}{d x} \sin{\left(6 x + 2 \right)}}{6} \]
   constant-multipleFactor out the constant 1/6.✓ Proved
4. \[ = 1 - \frac{\frac{d}{d x} \sin{\left(6 x + 2 \right)}}{6} \]
   derivativeThe derivative of x is 1.✓ Proved
5. \[ = - \frac{\cos{\left(6 x + 2 \right)} \frac{d}{d x} \left(6 x + 2\right)}{6} + 1 \]
   chainApply the chain rule to the sine function.✓ Proved
6. \[ = 1 - \cos{\left(6 x + 2 \right)} \]
   derivative algebraThe derivative of 6*x + 2 is 6. Simplify the expression by multiplying 1/6 and 6.✓ Proved