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

Problem 2.32 · hard

1. \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(4 x \right)}}{8}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(4 x \right)}}{8} \]
   sumApply the sum rule for differentiation.✓ Proved
3. \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(4 x \right)}}{8} \]
   constantPull out the constant factor from the second term.✓ Proved
4. \[ = - \frac{\cos{\left(4 x \right)} \frac{d}{d x} 4 x}{8} + \frac{d}{d x} \frac{x}{2} \]
   chainApply the chain rule to the sine term.✓ Proved
5. \[ = - \frac{\cos{\left(4 x \right)}}{2} + \frac{d}{d x} \frac{x}{2} \]
   derivative algebraDifferentiate the inner function 4*x. Simplify the constant coefficients.✓ Proved
6. \[ = \frac{1}{2} - \frac{\cos{\left(4 x \right)}}{2} \]
   derivativeEvaluate the derivative of the first term.✓ Proved
7. \[ = 0.5 - 0.5 \cos{\left(4 x \right)} \]
   simplifyFinal simplified form.✓ Proved