On 'Publishing' mathematical equations using MATLAB

I am at a loss to understand the publishing of mathematical equations using MATLAB for which you can use 'publish' command. I tried the following LATEX code in publish and got 'mixed' results. Some equations were recognized and published and some resulted in gibberish.

Also another problem I find is that inline Latex is misaligned (is raised above or below the adjacent text slightly) making it look abnormal.

May be I have to know better how to use this feature or may be MATLAB has to be improved!

%% 
% $$\frac{\mathrm{d}}{\mathrm{d}x}\arcsin x=\frac{1}{\sqrt{1-x^2}}$$
%% 
% Let $\arcsin( \frac{1}{2} \sqrt{2-\sqrt {2-x}})=y$ 
% $\implies-\dfrac\pi2\le y\le\dfrac\pi2$
% and $\dfrac12\sqrt{2-\sqrt {2-x}}=\sin y$
% $\implies2-\sqrt {2-x}=(2\sin y)^2$
% Using $\cos2A=2\cos^2A-1=1-2\sin^2A,$
% $\sqrt {2-x}=2-4\sin^2y=2\cos2y$
% $\implies x=2-(2\cos2y)^2=-2\cos4y$
% $\implies\cos4y=-\dfrac x2$
% $\arccos\left(-\dfrac x2\right)=\begin{cases}2\pi+4y  &\mbox{if }
% -2\pi\le4y<-\pi \\-4y &\mbox{if }  -\pi\le4y<0\\ 4y &\mbox{if } 0\le4y\le\pi \\ 2\pi-4y & \mbox{if }\pi<4y\le2\pi \end{cases}$

[Seetha Rama Raju Sanapala]

@Walter Roberson, Thanks for the editing. I am not confident of my LATEX knowledge, I copy and pasted some code from math.stackexchange. Some code was recognised, but some code resulted in error messages.