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

Problem 2.543 · hard

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