Derivative of \( \displaystyle \frac{x + 2}{x} \)

Problem 2.624 · medium

1. \[ \frac{d}{d x} \frac{x + 2}{x} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(1 + \frac{2}{x}\right) \]
   algebra simplifySplit the fraction into two parts. Simplify the first term.✓ Proved
3. \[ = \frac{d}{d x} 1 + \frac{d}{d x} \frac{2}{x} \]
   sumApply the sum rule.✓ Proved
4. \[ = \frac{d}{d x} \frac{2}{x} \]
   constant rewriteThe derivative of a constant is zero. Rewrite the term using a negative exponent.✓ Proved
5. \[ = 2 \frac{d}{d x} \frac{1}{x} \]
   constant-multiplePull out the constant factor.✓ Proved
6. \[ = - \frac{2}{x^{2}} \]
   power algebra simplifyApply the power rule. Simplify the expression. Rewrite with a positive exponent.✓ Proved