Is there a topological notion of the derivative?
There can be no purely topological definition of deriviative, because neither is the notion of differentiability preserved under homeomorphisms, nor (in cases where it happens to be preserved) does the derivative transform well under homeomorphisms (for instance the derivative could be nonzero before, and zero after application of a homeomorphism). General topology simply does not deal with notions of differentiation; you need a different category than topological spaces for that (for instance that of differentiable manifolds).
I am not sure if you count this as topological $X$ and I must admit that this might be unnecessarily "high-brow" but if your topological space is a locally ringed space (it's not that bad - just think of attaching a ring to every open set in some coherent way and when you "zoom" into a point $x$ you get a local ring $O_{X,x}$ - a ring with only one maximal ideal. Think manifolds or Euclidean space with rings of continuous $\mathbb{R}$-valued functions at each point), then one may define the (co)tangent space at $x$ as the vector space $m_x/m_x^2$ over the base field $O_{X,x}/m_x$ where $m_x$ is the unique maximal ideal of $O_{X,x}$.
The motivation of (the purely algebraic process of) quotienting out by the 2nd power of the ideal is exactly capturing the intuition of a derivative - you want to linearize everything in sight. This is what's done in algebraic geometry where the intuitive notion of smoothness is trickier and sometimes absent, but you still want to somehow have them anyway.
Hope that was at least fun!
I’m sorry .My last answer was not transferred properly .How can I send the mathematics symbols? However I am sending that again. I asked a similar question in www.artofproblemsolving.com, with topic name topological derivation definition. But I didn’t receive any answer. I’d like to explain more about this Idea. Indeed it seems that derivative and integral are concepts which are independent of coordinate system and maybe even independent of function concept. A geometric view of derivative is tangent. A tangent line of a curve at a point is a line which is very close to the curve at close points .This closeness can be interpreted by topological tools. On the other hand the integral of a function ∫abf(x)dx apparently depends on coordinate system, but by more inspection we see that coordinate system is only the boundary of integration .
Let’s denote a curve in a plane by the set cl(E) –E which E is open and is the greatest possible set in this respect. The following definition can be suggested for derivative: C= cl(E)-E , E is the greatest possible set. C is the curve. cl(NWEWV) -(NWEWV) = [cl(N)W(cl(V)-V)] [cl(V)W(cl(N)-N)] =( cl(N)W(cl(V))-( VWN) E,V and N are open .VT (a family of open sets) The important point in this way is the selection of type of V. If V be the open areas between two crossed line we get to ordinary derivative but we may select another sets. H= Vi (on all solutions of the above equation) S1, S2 / cl(V)-V= S1 S2 , S1WS2 ={x} x N D= cl(H)-H , H is the greatest possible set. D is the derivative. But the above definition is useless and can’t be used to combination of function that is very important.
Usually to define a derivative you need a smooth structure, because as Marc van Leeuwen points out in his answer, critical points are otherwise not preserved under homeomorphism.
I am submitting this answer just to point out the existence of a tangent microbundle in the category of topological manifolds. This may get closer to a "purely topological" notion of derivative.