Derivative of \( \displaystyle \frac{5 x}{2} - \frac{\sin{\left(10 x - 2 \right)}}{4} \)

Problem 2.974 · hard

1. \[ \frac{d}{d x} \left(\frac{5 x}{2} - \frac{\sin{\left(10 x - 2 \right)}}{4}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{d}{d x} \frac{\sin{\left(10 x - 2 \right)}}{4} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{\frac{d}{d x} \sin{\left(10 x - 2 \right)}}{4} \]
   constantPull out the constant factor 1/4.✓ Proved
4. \[ = \frac{5 \frac{d}{d x} x}{2} - \frac{\frac{d}{d x} \sin{\left(10 x - 2 \right)}}{4} \]
   constant-multiplePull out the constant factor 5/2.✓ Proved
5. \[ = \frac{5}{2} - \frac{\frac{d}{d x} \sin{\left(10 x - 2 \right)}}{4} \]
   derivativeDifferentiate x.✓ Proved
6. \[ = - \frac{\cos{\left(10 x - 2 \right)} \frac{d}{d x} \left(10 x - 2\right)}{4} + \frac{5}{2} \]
   trigApply the derivative of sine.✓ Proved
7. \[ = \frac{5}{2} - \frac{5 \cos{\left(10 x - 2 \right)}}{2} \]
   chain algebra simplifyApply the chain rule. Simplify the expression. Simplify the fraction.✓ Proved