# Notation: Functions of Derivatives of Variables

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:

$$f(x)=\frac{dx}{dt}=\dot{x}$$

$$f(\dot{x})=\dot{x}$$

where $$x=x(t)$$? This (possibly) creates a contradiction in that

$$f(x)=f(\dot{x})$$

• I'm not exactly sure what is your question; people write $\dot{x} = f(x)$ informally to mean the differential equation $\dot{x}(t) = f(x(t))$ for all $t$ on some domain. – Qi Zhu Oct 25 '19 at 17:22
• That's a good point. Here's an example. If I'm trying to define the force $f$ due to friction, I would say something like $f(\dot{x})=-k\dot{x}$. But I could also write $f(x)=-k\frac{dx}{dt}=-k\dot{x}$ couldn't I? – eball Oct 25 '19 at 18:29
• Well, the two equations you've written are different. The first takes $\dot{x}$ as the parameter, the second $x$. I think you probably mean the second equation. (Yes, you can write the second equation, here $f$ is a function of functions $x$ and you apply the differential operator on the right which is in itself a function.) – Qi Zhu Oct 25 '19 at 22:12
• @QiZhu, I see what you're saying. Would you agree that both $f(\dot{x})=-k\dot{x}$ and $f(x)=-k\frac{dx}{dt}$ are acceptable ways to write it, but maybe the second way is canonical? – eball Oct 28 '19 at 11:48
• No, I‘m saying that these two mean different things. – Qi Zhu Oct 28 '19 at 12:51

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.
• I think I meant "function that is a function of its variables derivative". Overall, I think what I'm asking is if it makes sense to write $f(x)=\dot{x}$. From your answer, it sounds like no, but I'm not quit sure why. – eball Oct 25 '19 at 17:11