What is the derivative of x*x^(1/2)?

Mar 18, 2016

$= \frac{3}{2} {x}^{\frac{1}{2}}$

Explanation:

$\frac{d}{\mathrm{dx}} \left(x \cdot {x}^{\frac{1}{2}}\right) = \frac{d}{\mathrm{dx}} \left({x}^{\frac{1}{2} + 1}\right) = \frac{d}{\mathrm{dx}} \left({x}^{\frac{3}{2}}\right) = \frac{3}{2} {x}^{\frac{3}{2} - 1} = \frac{3}{2} {x}^{\frac{1}{2}}$

Mar 18, 2016

$\frac{d}{d x} \left(x \cdot {x}^{\frac{1}{2}}\right) = \frac{3}{2} \cdot \sqrt{x}$
$\frac{d}{d x} \left(x \cdot {x}^{\frac{1}{2}}\right) = \frac{d}{d x} \left(x\right) \cdot {x}^{\frac{1}{2}} + \frac{d}{d x} \left({x}^{\frac{1}{2}}\right) \cdot x$
$\frac{d}{d x} \left(x \cdot {x}^{\frac{1}{2}}\right) = 1 \cdot {x}^{\frac{1}{2}} + \frac{1}{2} \cdot \left({x}^{\frac{1}{2} - 1}\right) \cdot x$
$\frac{d}{d x} \left(x \cdot {x}^{\frac{1}{2}}\right) = {x}^{\frac{1}{2}} + \frac{1}{2} \cdot {x}^{\frac{1}{2}}$
$\frac{d}{d x} \left(x \cdot {x}^{\frac{1}{2}}\right) = {x}^{\frac{1}{2}} \left(1 + \frac{1}{2}\right)$
$\frac{d}{d x} \left(x \cdot {x}^{\frac{1}{2}}\right) = \frac{3}{2} \cdot \sqrt{x}$