Derivative of \( \displaystyle \frac{3 x}{2} - \frac{3 \sin{\left(10 x + 2 \right)}}{20} \)

Problem 2.1107 · hard

1. \[ \frac{d}{d x} \left(\frac{3 x}{2} - \frac{3 \sin{\left(10 x + 2 \right)}}{20}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \frac{3 x}{2} - \frac{d}{d x} \frac{3 \sin{\left(10 x + 2 \right)}}{20} \]
   sum constant-multipleApply the difference rule. Factor out the constants.✓ Proved
3. \[ = \frac{3 \frac{d}{d x} x}{2} - \frac{3 \frac{d}{d x} \sin{\left(10 x + 2 \right)}}{20} \]
   constant-multipleFactor out the constants from each term.✓ Proved
4. \[ = \frac{3}{2} - \frac{3 \frac{d}{d x} \sin{\left(10 x + 2 \right)}}{20} \]
   derivative simplifyDifferentiate the first term. Simplify the first term.✓ Proved
5. \[ = - \frac{3 \cos{\left(10 x + 2 \right)} \frac{d}{d x} \left(10 x + 2\right)}{20} + \frac{3}{2} \]
   chainApply the chain rule to the sine function.✓ Proved
6. \[ = - \frac{3 \cos{\left(10 x + 2 \right)} \frac{d}{d x} x}{2} + \frac{3}{2} \]
   derivativeDifferentiate the inner function.✓ Proved
7. \[ = \frac{3}{2} - \frac{3 \cos{\left(10 x + 2 \right)}}{2} \]
   derivative simplify algebraDifferentiate the variable x. Simplify the derivative of x. Multiply 3/20 by 10.✓ Proved
8. \[ = \frac{3}{2} - 1.5 \cos{\left(10 x + 2 \right)} \]
   simplifyConvert the fraction to a decimal.✓ Proved