Derivative of \( \displaystyle \frac{5 x}{2} - \frac{5 \sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8} \)

Problem 2.1287 · hard

1. \[ \frac{d}{d x} \left(\frac{5 x}{2} - \frac{5 \sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{d}{d x} \frac{5 \sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{5 \frac{d}{d x} \sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8} \]
   constant-multipleFactor out the constant 5/8.✓ Proved
4. \[ = - \frac{5 \sin{\left(4 x \right)} \frac{d}{d x} \cos{\left(4 x \right)}}{8} - \frac{5 \cos{\left(4 x \right)} \frac{d}{d x} \sin{\left(4 x \right)}}{8} + \frac{d}{d x} \frac{5 x}{2} \]
   productApply the product rule to the second term.✓ Proved
5. \[ = \frac{5 \sin^{2}{\left(4 x \right)}}{2} - \frac{5 \cos^{2}{\left(4 x \right)}}{2} + \frac{d}{d x} \frac{5 x}{2} \]
   chain algebra constant-multiple algebraApply the chain rule to the trigonometric functions. Simplify the expression inside the parentheses. Factor out the 4. Simplify the constant coefficient.✓ Proved
6. \[ = \frac{5 \sin^{2}{\left(4 x \right)}}{2} - \frac{5 \cos^{2}{\left(4 x \right)}}{2} + \frac{5}{2} \]
   derivativeEvaluate the derivative of the first term.✓ Proved
7. \[ = \frac{5}{2} - \frac{5 \cos{\left(8 x \right)}}{2} \]
   simplifyUse the double angle identity cos(2u) = cos^2(u) - sin^2(u) and simplify.✓ Proved