$\mathscr{..}$ Expressions Fail to Render Properly in Org-Mode

Question

Consider the expression \mathscr{APPLE}. This should render as follows:

Yet in org-mode (using latex fragment previews), it renders as:

Moreover, when exporting, matscr renders properly in HTML (when viewed in a browser), but does not render properly when exporting to pdf! How does one fix this behavior?

EDIT: Certain unicode characters also fail to render properly. For example $\phi$ renders fine but $ϕ$ does not render fine. I hope these errors are related so that fixing one fixes the other.

EDIT2: I get the following error message when using the settings suggested by Omar: link.

EDIT3: The more specific error I'm getting (after checking *Org Preview LaTeX Output* buffer is:

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!
! Fatal fontspec error: "cannot-use-pdftex"
! 
! The fontspec package requires either XeTeX or LuaTeX.
! 
! You must change your typesetting engine to, e.g., "xelatex" or
! "lualatex"instead of plain "latex" or "pdflatex".
! 
! See the fontspec documentation for further information.
! 
! For immediate help type H <return>.
!...............................................  

Answer 1

The \mathscr problem is simply that Org mode LaTeX doesn't by default load the LaTeX package that provides that command. Just add #+LATEX_HEADER: \usepackage{mathrsfs} to your Org file and that should fix both preview and PDF export.

As for getting unicode math to work, you need, again, to load a package (so add #+LATEX_HEADER: \usepackage{unicode-math} to your Org file), but there's a wrinkle this time: the unicode-math package doesn´t work with pdflatex, only xetex or luatex, so you additionally need to customize the org-latex-pdf-process variable (for PDF export) and the org-preview-latex-process-alist and org-preview-latex-default-process variables (for LaTeX preview) to choose one of those compilers.