I’m not sure what the second part of your response is referring to. A change in notation does not necessarily imply any change to the amount of explanation given in a solution, surely?

So arcsin(x) (or commonly asin(x)) should be used rather than sin^(-1) (x).

Also, csc(x) (or perhaps cosec(x)) should be the convention and we should avoid any use of “sin (x) ^(-1)”.

The Maple interpretation of “sin (x+1)^5″ is unfortunate and you’re right, that would cause more confusion.

For example Maple interprets
sin (x+1)^5 as raising to power five the result of sin applied to x+1.

However in operator notation sin (x+1)^5 is interpretated as computing the sin of (x+1)^5.

Operator notation is undoubtably the notation of choice for the standart arithmetic operations such as *,+,^ etc
No one (except computer scientist in some cases) would write
+(1,*(2,3)) for 1+2*3.

However, my feeling is that the operator notation should be used only for elementary operations. If you start using unary operation notations like sin x, ln x, after introducing sin and ln as functions and you did not give an explanation that operators are associated with functions that can be expressed by a particular notation (which would go too far at low level maths) then you are potentially confusing students.

And then you can confuse them even more when you consider composition: Does sin^(-1)x means arcsin(x) or 1/sin(x)?

If you only use pure functional notation then
sin^(-1) (x) means arcsin(x), sin (x) ^(-1) means 1/sin(x), and sin (x^(-1)) means sin(1/x).

Linear functions can be distributed out of their parentheses. This is why you can write int(x+y) = int(x)+int(y) or laplace(x+y)=laplace(x)+laplace(y). However, it’s critical to math literacy to recognize that one of the things you have to know about every function is whether or not it’s a linear operation on the argument. The four basic operations are all linear–but when you cross into college nobody informs you of when that stops happening. That’s the issue.

I also like subscript notation for derivatives, but as you just found, it is a bother to type it on the Web.

For reference, you can type your

f[subscript g][multiplication dot]g[subscript x](x)

as

f_g·g_x(x)

and it will look like the following, as you intended:

f_g·g_x(x)

As for derivative symbols, a neat alternative I found is to use subscript of the variable that the derivative is with respect to. So, for a function f(x) (I’d have made the f in script style if I could), its derivative would be fx(x) (I’d have made the first x subscript on the f here if I could). The chain rule can be written thusly: given y=f(g(x)), the derivative of y with respect to x could be written f[subscript g][multiplication dot]g[subscript x](x). Anyway it’s more of an exercise for my own satisfaction to make concisely written notes, as students would be learning the other notations for decades to come I imagine.

I’m definately going to make use of this in the future.