# How do you find the derivative of lnsqrt x?

Apr 9, 2015

In general
$\frac{d \ln \left(a\right)}{\mathrm{da}} = \frac{1}{a}$

Specifically if $a = \sqrt{x}$
$\frac{d \ln \left(\sqrt{x}\right)}{d \sqrt{x}} = \frac{1}{\sqrt{x}}$

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

$\frac{d \sqrt{x}}{\mathrm{dx}} = \frac{1}{2} {x}^{- \frac{1}{2}} = \frac{1}{2 \sqrt{x}}$

$\frac{d \ln \left(\sqrt{x}\right)}{\mathrm{dx}} = \frac{d \ln \left(\sqrt{x}\right)}{d \sqrt{x}} \cdot \frac{d \sqrt{x}}{\mathrm{dx}}$

$= \frac{1}{\sqrt{x}} \cdot \frac{1}{2 \sqrt{x}}$

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

That is, the derivative of $\ln \sqrt{x}$
is
$\frac{1}{2 x}$