Derivative of \( \displaystyle \frac{x + 1}{\sqrt{x}} \)

Problem 2.317 · medium

1. \[ \frac{d}{d x} \frac{x + 1}{\sqrt{x}} \]
   derivative rewriteStart with the derivative of the function. Rewrite the square root using a fractional exponent.✓ Proved
2. \[ = \frac{d}{d x} \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right) \]
   algebra algebraDistribute the term x**(-1/2) into the parentheses. Simplify the exponents using exponent laws.✓ Proved
3. \[ = \frac{d}{d x} x^{-0.5} + \frac{d}{d x} x^{0.5} \]
   sumApply the sum rule for derivatives.✓ Proved
4. \[ = - \frac{0.5}{x^{1.5}} + \frac{0.5}{x^{0.5}} \]
   powerApply the power rule to each term.✓ Proved
5. \[ = - \frac{0.5}{x^{1.5}} + \frac{0.5}{\sqrt{x}} \]
   rewriteRewrite negative exponents as fractions.✓ Proved
6. \[ = \frac{0.5}{\sqrt{x}} - \frac{0.5}{x^{\frac{3}{2}}} \]
   algebraSimplify x**1.5 as x*sqrt(x).✓ Proved
7. \[ = \frac{0.5 x - 0.5}{x^{\frac{3}{2}}} \]
   algebraFind a common denominator.✓ Proved
8. \[ = \frac{x - 1}{2 x^{\frac{3}{2}}} \]
   simplifySimplify the final expression.✓ Proved