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

Problem 2.1369 · hard

1. \[ \frac{d}{d x} \left(\frac{3 x}{2} - \frac{3 \sin{\left(10 x - 2 \right)}}{20}\right) \]
   Start 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 sum 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-multipleApply the constant multiple rule to both terms.✓ Proved
4. \[ = \frac{3}{2} - \frac{3 \frac{d}{d x} \sin{\left(10 x - 2 \right)}}{20} \]
   derivative simplifyDifferentiate x. Simplify the first term.✓ Proved
5. \[ = - \frac{3 \cos{\left(10 x - 2 \right)} \frac{d}{d x} \left(10 x - 2\right)}{20} + \frac{3}{2} \]
   trigDifferentiate the sine function.✓ Proved
6. \[ = - \frac{3 \cos{\left(10 x - 2 \right)} \frac{d}{d x} x}{2} + \frac{3}{2} \]
   chainApply the chain rule.✓ Proved
7. \[ = \frac{3}{2} - \frac{3 \cos{\left(10 x - 2 \right)}}{2} \]
   derivative simplify algebraDifferentiate the inner function 10*x - 2. Simplify the derivative of the inner function. Multiply 3/20 by 10.✓ Proved
8. \[ = \frac{3}{2} - 1.5 \cos{\left(10 x - 2 \right)} \]
   simplifyFinal simplification.✓ Proved