Derivative of \( \displaystyle \frac{5 x}{2} - \frac{5 \sin{\left(6 x + 2 \right)}}{12} \)

Problem 2.969 · hard

1. \[ \frac{d}{d x} \left(\frac{5 x}{2} - \frac{5 \sin{\left(6 x + 2 \right)}}{12}\right) \]
   Differentiate the function with respect to x.✓ Proved
2. \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{d}{d x} \frac{5 \sin{\left(6 x + 2 \right)}}{12} \]
   sum constant-multipleApply the sum rule. Factor out the constants.✓ Proved
3. \[ = \frac{5 \frac{d}{d x} x}{2} - \frac{5 \frac{d}{d x} \sin{\left(6 x + 2 \right)}}{12} \]
   constant-multipleSeparate the constants from the derivatives.✓ Proved
4. \[ = \frac{5}{2} - \frac{5 \frac{d}{d x} \sin{\left(6 x + 2 \right)}}{12} \]
   derivative algebraDifferentiate x. Simplify the first term.✓ Proved
5. \[ = - \frac{5 \cos{\left(6 x + 2 \right)} \frac{d}{d x} \left(6 x + 2\right)}{12} + \frac{5}{2} \]
   chainApply the chain rule to the sine function.✓ Proved
6. \[ = \frac{5}{2} - \frac{5 \cos{\left(6 x + 2 \right)}}{2} \]
   derivative algebra simplifyDifferentiate the inner function 6*x + 2. Multiply the constants. Simplify the fraction 30/12.✓ Proved