Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(4 x + 4 \right)}}{8} \)

Problem 2.61 · hard

1. \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(4 x + 4 \right)}}{8}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(4 x + 4 \right)}}{8} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(4 x + 4 \right)}}{8} \]
   constantFactor out the constant 1/8.✓ Proved
4. \[ = \frac{1}{2} - \frac{\frac{d}{d x} \sin{\left(4 x + 4 \right)}}{8} \]
   derivativeDifferentiate the first term.✓ Proved
5. \[ = - \frac{\cos{\left(4 x + 4 \right)} \frac{d}{d x} \left(4 x + 4\right)}{8} + \frac{1}{2} \]
   chainApply the chain rule to the sine term.✓ Proved
6. \[ = \frac{1}{2} - \frac{\cos{\left(4 x + 4 \right)}}{2} \]
   derivative algebra simplifyDifferentiate the inner function 4*x + 4. Multiply the terms. Simplify the fraction.✓ Proved