Calculus: Early Transcendentals, Chapter 3, 3.1, Section 3.1, Problem 22

Differentiate the function $y = \sqrt{x}(x - 1)$


$
\begin{equation}
\begin{aligned}
\text{We have } y &= x^{\frac{1}{2}} (x - 1)\\
\\
y &= x^{\frac{1}{2} + 1} - x^{\frac{1}{2}} = x^{\frac{3}{2}} - x^{\frac{1}{2}}
\end{aligned}
\end{equation}
$


So,

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