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

Problem 2.1177 · hard

1. \[ \frac{d}{d x} \left(- \frac{x}{2} + \frac{\sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \frac{\sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{\frac{d}{d x} \sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8} \]
   constant-multipleFactor out the constant.✓ Proved
4. \[ = \frac{\sin{\left(4 x \right)} \frac{d}{d x} \cos{\left(4 x \right)}}{8} + \frac{\cos{\left(4 x \right)} \frac{d}{d x} \sin{\left(4 x \right)}}{8} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]
   productApply the product rule.✓ Proved
5. \[ = - \frac{\sin^{2}{\left(4 x \right)}}{2} + \frac{\cos^{2}{\left(4 x \right)}}{2} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]
   chain algebra constant-multipleApply the chain rule to the trigonometric terms. Simplify the expression inside the parentheses. Factor out the 4.✓ Proved
6. \[ = - \frac{\sin^{2}{\left(4 x \right)}}{2} + \frac{\cos^{2}{\left(4 x \right)}}{2} - \frac{1}{2} \]
   derivativeEvaluate the derivative of the first term.✓ Proved
7. \[ = \frac{\cos{\left(8 x \right)}}{2} - \frac{1}{2} \]
   rewrite algebra simplifyUse the double angle identity cos(2*theta) = cos(theta)**2 - sin(theta)**2. Simplify the argument of the cosine function. Final simplified form.✓ Proved