Are there any mathematical expressions which have no explicit differentials?

Question

We all know how often it happens that there exists no explicit indefinite integral for a given mathematical expression, but how about the other way around with differentials?

Alexander, you've made a good argument that any expression containing already differentiable functions is itself also differentiable. But are there any non-pathological functions for which there aren't any explicit differentials? Any obscure functions out there, like the Mathieu functions? But even these have explicit differentials.

Answer 1

No.

Lets define 'expression'.

A set of all exprerssionable functions E is defined
(by me, for the purpopse of this problem) as any subset
of the set of all functions F, which is generated from subset
G of primitive functions by the following 5 primitive
operations [+]/[-]/[divide]/[multiply]/[composite [&]fg = f(g(x))].
Invertion f^(-1) is not here.

Then, for example < ' means operator of differentiation>

'[+](f,g) = [+](f', g') = [+]' -> '[+](,) = [+](',')

'[multiply](,) = [+]([multiply](',), (,'))

[&](f,g) = [multiply]([&](',),')

So if all members g of G have g' in E, then all elements e
of E also have e' in E.


For integration it obviously does not work because, for
example [integrate][multiply] is not defined in terms of the
five primitive operations and [integrate].

Answer 2

Yes, as a function might not be differentiable anywhere:
EXAMPLE x=5 dy/dx is not defined.

Also some piecewise functions:
f(x)=1 for rationals, but 0 for irrationals.. has no explicit derivative.

but, in general, a "normal" function does.. we can evaluate the limit as x=>0 [f(x+h)-f(x)] / h for most fumctions the limit will exist.

good luck!