Here's the Taylor series of expansion about

where the terms of order $2$ and higher have been discarded to create a linear approximation to $.$ This operation is easy for basic functions such as $.$ When is more complicated, such as

it gets tedious. Here we'll use MATLAB symbolic to ease the pain.

Using the Learner Template linearize above about The template has an example for about $.$ You can follow that workflow. Be sure to name your function g$($x$)=...$ and your linearized version gLin.syms x dx x$0$

y$($x$)=$ sqrt$($x$)$;

yLin $=$ taylor$($y$,$x$,$'ExpansionPoint',x$0,$'Order',$2)$

$\%$ Since we've defined x $=$ x$0+$ dx$,$ we need to substitute x accordingly

yLin $=$ subs$($yLin$,$x$,$x$0+$dx$)$

$\%$ Now lets set x$0$ to the value we are interested in$,$ x$0=1$

yLin $=$ subs$($yLin$,$x$0,1)$

g$($x$)=($x$^2)*$log$($x$)$