The idea of a total derivative is to find a linear function that is the tangent in a point, so to say. This is something I somehow can imagine.
Do we have such an idea for partial derivatives to? That is, can we "draw" it as some kind of tangent at a point to? I think rather not (how should we draw a tangent only for one coordinate direction)
Partial derivatives occur in the context of a function of multiple, say $n,$ variables. The graph of that function is a subset of $R^{n+1},$ typically a hypersurface if the function is well-behaved. The partial derivative is the slope of a tangent line (that's right, a one-dimensional subspace of an ($n+1$)-dimensional space) to the hypersurface.
For a function of two variables (the only case we can really draw) the graph is a surface in 3 dimensions. The partial derivative with respect to $x$ is the slope of a line tangent to that surface and parallel to the ($x$, $z$) -plane.