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

Problem 2.111 · hard

1. \[ \frac{d}{d x} \left(x - \frac{\sin{\left(10 x \right)}}{10}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} x - \frac{d}{d x} \frac{\sin{\left(10 x \right)}}{10} \]
   sumApply the sum rule for differentiation.✓ Proved
3. \[ = \frac{d}{d x} x - \frac{\frac{d}{d x} \sin{\left(10 x \right)}}{10} \]
   constant-multipleFactor out the constant 1/10.✓ Proved
4. \[ = 1 - \frac{\frac{d}{d x} \sin{\left(10 x \right)}}{10} \]
   derivativeThe derivative of x is 1.✓ Proved
5. \[ = - \frac{\cos{\left(10 x \right)} \frac{d}{d x} 10 x}{10} + 1 \]
   chainApply the chain rule to sin(10*x).✓ Proved
6. \[ = 1 - \cos{\left(10 x \right)} \]
   derivative algebraThe derivative of 10*x is 10. Simplify the expression by multiplying the constants.✓ Proved