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

Problem 2.358 · hard Beautiful

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. \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8} \]
   constant-multipleFactor out the constant.✓ Proved
4. \[ = - \frac{\sin{\left(4 x \right)} \frac{d}{d x} \cos{\left(4 x \right)}}{8} - \frac{\cos{\left(4 x \right)} \frac{d}{d x} \sin{\left(4 x \right)}}{8} + \frac{d}{d x} \frac{x}{2} \]
   productApply the product rule.✓ Proved
5. \[ = \frac{\sin^{2}{\left(4 x \right)}}{2} - \frac{\cos^{2}{\left(4 x \right)}}{2} + \frac{d}{d x} \frac{x}{2} \]
   chain algebraApply the chain rule to the trigonometric terms. Simplify the expression inside the parentheses.✓ Proved
6. \[ = \frac{\sin^{2}{\left(4 x \right)}}{2} - \frac{\cos^{2}{\left(4 x \right)}}{2} + \frac{1}{2} \]
   derivative constant-multiple algebraEvaluate the derivative of x/2. Distribute the 1/8. Factor out 1/2.✓ Proved
7. \[ = \frac{1}{2} - \frac{\cos{\left(8 x \right)}}{2} \]
   trig algebra algebraUse the double angle identity cos(2*theta) = cos^2(theta) - sin^2(theta). Simplify the argument of the cosine function. Factor out 1/2.✓ Proved
8. \[ = \sin^{2}{\left(4 x \right)} \]
   simplifyUse the identity 1 - cos(2*theta) = 2*sin(theta)**2 with theta = 4*x.✓ Proved