Question

1. (5 points) differentiate ( h(x) = sqrt{x}(x - 2) ).

Explanation:

Step1: Rewrite in exponential form

$h(x) = x^{1/2}(x - 2)$

Step2: Apply product rule

Let $u = x^{1/2}$, $v = x - 2$. Then $u' = \frac{1}{2}x^{-1/2}$, $v' = 1$.
Product rule: $h'(x) = u'v + uv'$
$h'(x) = \frac{1}{2}x^{-1/2}(x - 2) + x^{1/2}(1)$

Step3: Simplify the expression

$\frac{1}{2}x^{-1/2}(x - 2) = \frac{x - 2}{2\sqrt{x}} = \frac{\sqrt{x}}{2} - \frac{1}{\sqrt{x}}$
$x^{1/2} = \sqrt{x}$
Combine terms: $\frac{\sqrt{x}}{2} - \frac{1}{\sqrt{x}} + \sqrt{x} = \frac{3\sqrt{x}}{2} - \frac{1}{\sqrt{x}}$

Step4: Rewrite with common denominator (optional)

$\frac{3x - 2}{2\sqrt{x}}$

Answer:

$\frac{3\sqrt{x}}{2} - \frac{1}{\sqrt{x}}$ (or equivalent form $\frac{3x - 2}{2\sqrt{x}}$)