Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(10 x - 6 \right)}}{20} \)

Problem 2.550 · hard

1. \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(10 x - 6 \right)}}{20}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(10 x - 6 \right)}}{20} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(10 x - 6 \right)}}{20} \]
   constantPull out the constant factor.✓ Proved
4. \[ = - \frac{\cos{\left(10 x - 6 \right)} \frac{d}{d x} \left(10 x - 6\right)}{20} + \frac{d}{d x} \frac{x}{2} \]
   chainApply the chain rule to the sine function.✓ Proved
5. \[ = - \frac{\left(- \frac{d}{d x} 6 + \frac{d}{d x} 10 x\right) \cos{\left(10 x - 6 \right)}}{20} + \frac{d}{d x} \frac{x}{2} \]
   sumDifferentiate the inner function.✓ Proved
6. \[ = - \frac{\cos{\left(10 x - 6 \right)} \frac{d}{d x} 10 x}{20} + \frac{d}{d x} \frac{x}{2} \]
   constantThe derivative of the constant 6 is 0.✓ Proved
7. \[ = - \frac{\cos{\left(10 x - 6 \right)} \frac{d}{d x} x}{2} + \frac{d}{d x} \frac{x}{2} \]
   constant algebra simplifyThe derivative of 10*x is 10. Multiply the constants. Simplify the fraction 10/20.✓ Proved
8. \[ = \frac{1}{2} - \frac{\cos{\left(10 x - 6 \right)}}{2} \]
   derivative simplifyEvaluate the derivatives of the remaining terms. Simplify the final expression.✓ Proved