Calculus: Early Transcendentals, Chapter 3, 3.1, Section 3.1, Problem 24

Differentiate the function $\displaystyle \frac{x^2 - 2\sqrt{x}}{x}$


$\begin{equation} \begin{aligned} \text{We have } y &= \frac{x^2 - 2x^{\frac{1}{2}}}{x} \\ \\ &= \frac{x^2}{x} - \frac{2x^{\frac{1}{2}}}{x}\\ \\ &= x^{2 - 1} - 2x^{\frac{1}{2} - 1}\\ \\ &= x - 2x^{-\frac{1}{2}} \end{aligned} \end{equation}$


So,

$\begin{equation} \begin{aligned} \frac{dy}{dx} &= \frac{d}{dx} \left( x - 2x^{-\frac{1}{2}} \right)\\ \\ &= \frac{d}{dx} (x) - 2 \cdot \frac{d}{dx} \left( x^{-\frac{1}{2}} \right)\\ \\ &= (1) - 2 \cdot \left( - \frac{1}{2} \right) x^{-\frac{1}{2} - 1}\\ \\ &= 1 + x^{-\frac{3}{2}} \text{ or } 1 + \frac{1}{x^{\frac{3}{2}}} \end{aligned} \end{equation}$