I have some doubts regarding function notation:
First If I present a function I write:$f(x)$
If I write it's inverse:$f^{-1}(x)$
So why doesn't$f(f(x))=f^2(x)$
Second If $\frac{df(x)}{dx}=f'(x)$ and $\frac{d^2f(x)}{dx^2}=f''(x)$.
So how do you write probably $\frac{d^nf(x)}{dx^n}=f'(x)$.
Does like this:$f'''^{\cdots\text{n times}}(x)$?
Third What difference between $f^n(x),f(x)^n\text{ and }(f(x))^n$
On
It depends on the context and the conventions. Four your second question, people usually write $f^{(n)}$. For your first question, some people actually use $f^2$ as an abbreviation for $f \circ f$, but it's not that common.
On
First: The inverse function notation has nothing to do with exponentiation, therefore $f(f(x))$ is not written as $f^2(x)$. Note, that sometimes on calculators you might see $\cos^{-1}{}$ which means $\arccos$, NOT $\frac{1}{\cos}$.
Second: For derivatives of order $n \geq 4$, typical notation is $f^{(n)}(x)$
Third: This I am not entirely sure of, but to avoid any confusion, I'd only use $(f(x))^n$.
On
When writing $f^2(x)$, it normally refers to $f(f(x)$. In general: $$f^n(x) = \overbrace{f(f(\ldots f}^{n \text{ times}}(x))\ldots)\\f(x)^n=(f(x))^n\text{ is $f(x)$ raised to the $n$th power}$$
You can certainly write $\dfrac{d^nf(x)}{dx}=f''^\ldots(x)$. In general, one would write $\dfrac{d^nf(x)}{dx}=f^{(n)}(x)$.
Although, be aware that the context will often dictate what $f^n(x)$ is.
On
I hope this helps:
$$f^n(x) = f(f^{n-1}(x)) = \underbrace{f(f(f(\cdots f(x))\cdots)}_{n \text{ times}} = \underbrace{(f^\circ f^\circ f^\circ \cdots f)}_{n \text{ times}}(x)$$
$$\frac{d^n}{dx^n}f(x) = f^{(n)}(x)$$
$$(f(x))^n = \underbrace{f(x)·f(x)\ldots f(x)}_{n \text{ times}}$$
As far as I know, there's no such thing as $f(x)^n$ but it might mean $(f(x))^n$
This is indeed ambiguous. Unfortunately $f^n$ is sometimes used to refer to the $n$-th iterate of a function and other times is commonly used to refer to the $n$-th exponentiation. The same notation was adopted by different branches of mathematics to mean different things and both persist in using it so. Thus when ever you see $f^2$, you should check the context to determine if the author meant the composition $f\circ f$ or the product $f\cdot f$.
More confusingly, the convention that $f^{-1}$ means the iterate inverse of a function has become the standard even when $f^n$ is otherwise used for exponentiation.
Some mathematicians adopt the notation $f^{\circ n}$ to be clear that they mean the $n$ iterate.
We never go passed three primes. The convention is that $f^{(n)}$ means the $n$-th derivative of a function (with respect to its argument).
Clarity. $f^n(x)$ is ambiguous; it may mean $f\circ f^{n-1}(x)$ or commonly $f\cdot f^{n-1}$. It is possible that $f(x)^n$ could be parsed as $f(x^n)$. However, $(f(x))^n$ is fairly unambiguous; although it is more chunky.