I'm sorry if this has been asked before and if the title is unclear; I'm not sure how to search for this question. Is it proper to write a function that is a function it's variables derivative as:
$$f(x)=\frac{dx}{dt}=\dot{x}$$
instead of
$$f(\dot{x})=\dot{x}$$
where $x=x(t)$? This (possibly) creates a contradiction in that
$$f(x)=f(\dot{x})$$
Thanks in advance!
I cannot imagine a function that is a function of its own derivative. Sounds like a circular reference.
Perhaps you mean this. Consider a function $x = x(t)$ and you can have a function of both $x(t)$ and $x'(t)$ at the same time? For example, consider an iteration of Newton's method, which computes $$ N(t, x(t), x'(t)) = t - \frac{x(t)}{x'(t)}. $$ This is well-defined as a map $N : \mathbb{R}^3 \to \mathbb{R}$.
If you want to map functions to other functions, it's a different story. For example, consider the space $\mathcal{S}$ of all one-variable functions, differentiable infinitely many times on $[0,1]$, e.g. $\cos(x)$ or $e^x$ or any polynomial. Then you can have a map $D : \mathcal{S} \to \mathcal{S}$ defined by $$ D[f(x)] = f'(x), $$ thus mapping $e^x \to e^x$ and $\sin x \to \cos x$, for example.