Derivative of \( \displaystyle \frac{\cos{\left(x \right)}}{x} \)

Problem 2.409 · hard

1. \[ \frac{d}{d x} \frac{\cos{\left(x \right)}}{x} \]
   derivative algebraStart with the derivative of the function. Rewrite the quotient as a product.✓ Proved
2. \[ = \cos{\left(x \right)} \frac{d}{d x} \frac{1}{x} + \frac{\frac{d}{d x} \cos{\left(x \right)}}{x} \]
   productApply the product rule.✓ Proved
3. \[ = \cos{\left(x \right)} \frac{d}{d x} \frac{1}{x} - \frac{\sin{\left(x \right)}}{x} \]
   trigDifferentiate cos(x).✓ Proved
4. \[ = - \frac{\sin{\left(x \right)}}{x} - \frac{\cos{\left(x \right)}}{x^{2}} \]
   power algebraDifferentiate 1/x using the power rule. Simplify the terms.✓ Proved
5. \[ = \frac{- x \sin{\left(x \right)} - \cos{\left(x \right)}}{x^{2}} \]
   simplifyCombine into a single fraction.✓ Proved