Derivative of \( \displaystyle \frac{5 x}{2} - \frac{5 \sin{\left(4 x - 6 \right)}}{8} \)

Problem 2.1210 · hard

1. \[ \frac{d}{d x} \left(\frac{5 x}{2} - \frac{5 \sin{\left(4 x - 6 \right)}}{8}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{d}{d x} \frac{5 \sin{\left(4 x - 6 \right)}}{8} \]
   sum constant-multipleApply the sum rule. Factor out the constants.✓ Proved
3. \[ = \frac{5 \frac{d}{d x} x}{2} - \frac{5 \frac{d}{d x} \sin{\left(4 x - 6 \right)}}{8} \]
   constant-multipleApply the constant multiple rule to both terms.✓ Proved
4. \[ = \frac{5}{2} - \frac{5 \frac{d}{d x} \sin{\left(4 x - 6 \right)}}{8} \]
   derivative algebraDifferentiate x. Simplify the first term.✓ Proved
5. \[ = - \frac{5 \cos{\left(4 x - 6 \right)} \frac{d}{d x} \left(4 x - 6\right)}{8} + \frac{5}{2} \]
   trigApply the derivative of sine.✓ Proved
6. \[ = - \frac{5 \cos{\left(4 x - 6 \right)} \frac{d}{d x} \left(-6\right)}{8} - \frac{5 \cos{\left(4 x - 6 \right)} \frac{d}{d x} 4 x}{8} + \frac{5}{2} \]
   sumApply the sum rule inside the chain rule.✓ Proved
7. \[ = \frac{5}{2} - \frac{5 \cos{\left(4 x - 6 \right)}}{2} \]
   derivative constant algebraDifferentiate the terms inside the chain rule. The derivative of the constant -6 is 0. Multiply 5/8 by 4.✓ Proved