[cygx]

Note that in context of differential geometry, the name in the denominator is not associated with the function, but a coordinate system on its domain:

Let M be a differential manifold, eg M = ℝ² and φ a chart, eg cartesian coordinates φ: p ↦(x(p), y(p)) where x,y: M → ℝ. Then, ∂/∂x denotes the holonomic vector field tangent to the coordinate lines t ↦ φ⁻¹(x(p) + t, y(p)) through any p ∈ M.

It is convenient to identify vectors and directional derivatives (this is in fact one possible way to define tangent vectors on manifolds), which for a function f: M → ℝ yields

(∂f/∂x)(p) = ∂/∂x|ₚ f = lim_{h → 0} ( f(φ⁻¹(x(p) + h, y(p))) - f(p) ) / h

[bikenaga]

This is a good point. Noting that "d/dx" is a vector (field) is an antidote to the ideas that vectors are little arrows, or "a magnitude and a direction", or an ordered list with numbers in the slots. Vectors can be thought of in lots of ways, and here they're thought of as operators on functions.