Derivative of \( \displaystyle - \frac{x}{2} + \frac{\sin{\left(10 x - 2 \right)}}{20} \)

Problem 2.193 · hard

1. \[ \frac{d}{d x} \left(- \frac{x}{2} + \frac{\sin{\left(10 x - 2 \right)}}{20}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \frac{\sin{\left(10 x - 2 \right)}}{20} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{\frac{d}{d x} \sin{\left(10 x - 2 \right)}}{20} \]
   constant-multipleFactor out the constant 1/20.✓ Proved
4. \[ = \frac{\cos{\left(10 x - 2 \right)} \frac{d}{d x} \left(10 x - 2\right)}{20} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]
   chainApply the chain rule to the sine term.✓ Proved
5. \[ = \frac{\cos{\left(10 x - 2 \right)}}{2} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]
   derivativeDifferentiate the inner function 10*x - 2.✓ Proved
6. \[ = \frac{\cos{\left(10 x - 2 \right)}}{2} - \frac{1}{2} \]
   derivative simplifyDifferentiate the first term -x/2. Simplify the fraction 10/20.✓ Proved