Here are a few more examples:

Type this See that Comment
\`x^2+y_1+z_12^34\` `x^2+y_1+z_12^34` subscripts as in TeX, but numbers are treated as a unit
\`sin^-1(x)\` `sin^-1(x)` function names are treated as constants
\`d/dxf(x)=lim_(h->0)(f(x+h)-f(x))/h\` `d/dxf(x)=lim_(h->0)(f(x+h)-f(x))/h` complex subscripts are bracketed, displayed under lim
\$\frac{d}{dx}f(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\$ $\frac{d}{dx}f(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$ standard LaTeX notation is an alternative
\`f(x)=sum_(n=0)^oo(f^((n))(a))/(n!)(x-a)^n\` `f(x)=sum_(n=0)^oo(f^((n))(a))/(n!)(x-a)^n` f^((n))(a) must be bracketed, else the numerator is only `a`
\$f(x)=\sum_{n=0}^\infty\frac{f^{(n)}(a)}{n!}(x-a)^n\$ $f(x)=\sum_{n=0}^\infty\frac{f^{(n)}(a)}{n!}(x-a)^n$ standard LaTeX produces the same result
\`int_0^1f(x)dx\` `int_0^1f(x)dx` subscripts must come before superscripts
\`[[a,b],[c,d]]((n),(k))\` `[[a,b],[c,d]]((n),(k))` matrices and column vectors are simple to type
\`x/x={(1,if x!=0),(text{undefined},if x=0):}\` `x/x={(1,if x!=0),(text{undefined},if x=0):}` piecewise defined function are based on matrix notation
\`a//b\` `a//b` use // for inline fractions
\`(a/b)/(c/d)\` `(a/b)/(c/d)` with brackets, multiple fraction work as expected
\`a/b/c/d\` `a/b/c/d` without brackets the parser chooses this particular expression
\`((a*b))/c\` `((a*b))/c` only one level of brackets is removed; * gives standard product
\`sqrt sqrt root3x\` `sqrt sqrt root3x` spaces are optional, only serve to split strings that should not match
\`<< a,b >> and {:(x,y),(u,v):}\` `<< a,b >> and {:(x,y),(u,v):}` angle brackets and invisible brackets
\`(a,b]={x in RR | a < x <= b}\` `(a,b]={x in RR | a < x <= b}` grouping brackets don't have to match
\`abc-123.45^-1.1\` `abc-123.45^-1.1` non-tokens are split into single characters,
but decimal numbers are parsed with possible sign
\`hat(ab) bar(xy) ulA vec v dotx ddot y\` `hat(ab) bar(xy) ulA vec v dotx ddot y` accents can be used on any expression (work well in IE)
\`bb{AB3}.bbb(AB].cc(AB).fr{AB}.tt[AB].sf(AB)\` `bb{AB3}.bbb(AB].cc(AB).fr{AB}.tt[AB].sf(AB)` font commands; can use any brackets around argument
\`stackrel"def"= or \stackrel{\Delta}{=}" "("or ":=)\` `stackrel"def"= or \stackrel{\Delta}{=}" "("or ":=)` symbols can be stacked
\`{::}_(\ 92)^238U\` `{::}_(\ 92)^238U` prescripts simulated by subsuperscripts