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!