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

Problem 2.2070 · hard Beautiful

1. \[ \frac{d}{d x} \left(x - \frac{\sin{\left(4 x \right)} \cos{\left(4 x \right)}}{4}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} x - \frac{d}{d x} \frac{\sin{\left(4 x \right)} \cos{\left(4 x \right)}}{4} \]
   sumApply the sum rule.✓ Proved
3. \[ = 1 - \frac{d}{d x} \frac{\sin{\left(4 x \right)} \cos{\left(4 x \right)}}{4} \]
   constantDifferentiate the first term.✓ Proved
4. \[ = 1 - \frac{\frac{d}{d x} \sin{\left(4 x \right)} \cos{\left(4 x \right)}}{4} \]
   constant-multipleFactor out the constant 1/4.✓ Proved
5. \[ = - \frac{\sin{\left(4 x \right)} \frac{d}{d x} \cos{\left(4 x \right)}}{4} - \frac{\cos{\left(4 x \right)} \frac{d}{d x} \sin{\left(4 x \right)}}{4} + 1 \]
   productApply the product rule.✓ Proved
6. \[ = \sin^{2}{\left(4 x \right)} - \cos^{2}{\left(4 x \right)} + 1 \]
   chain algebra simplifyApply the chain rule to both terms. Simplify the terms inside the parentheses. Distribute the 1/4.✓ Proved
7. \[ = 1 - \cos{\left(8 x \right)} \]
   rewrite simplify algebra simplifyUse the double angle identity cos(2*theta) = cos^2(theta) - sin^2(theta) with theta = 4x. Wait, the identity is cos(2*theta) = cos^2(theta) - sin^2(theta). Let's re-evaluate the identity step. The expression is 1 - cos(2*theta) where theta is 4x. Final simplification.✓ Proved