In Clarke's Primer on Tensor Calculus (online .pdf),
Equation (14) is $$ \frac{\partial x^i}{\partial \tilde{x}^k} \frac{\partial \tilde{x}^k}{\partial x^j} = \delta^i_j = \frac{\partial x^i}{\partial x^j} $$
My question (surely embarassing) is about the part on the right, $\frac{\partial x^i}{\partial x^j} = \delta^i_j$.
$x^i$ denotes a "point", with i indexing a coordinate or dimension from 1 to 3 in the 3d case. Then the derivative expression $\frac{\partial x^i}{\partial x^j}$ is (I believe) about how dimension i of point x changes as dimension j of the same point changes.
I fear i do not understand what this even means at the most basic level. If the point is moving, its coordinates change, but how they change depends on both the direction of motion, and whether the directions i,j are orthogonal. In an orthogonal case, one can move "on the diagonal", simultaneously changing each coordinate of x, so the \delta relation will not be correct. If the directions i,j are not orthogonal, moving in direction i will result in $x^j$ changing.
By "direction i", i mean the direction resulting when one coordinate ( a particular value of i) is changes and the other two are held fixed. I think this is denoted $\partial_i x$.
I will try to give some intuition.
The change in coordinate $i$ (or really of any function $f$) as coordinate $j$ changes is not well defined: As you rightly observe, two ways of moving a point may very well result in different changes of $f$ even if coordinate $j$ behaves the same.
However, that is not how $\frac{\partial f}{\partial x^j}$ is defined. Instead, it is the change in $f$ when $x^j$ changes and all other coordinates don’t change and this is a well-defined notion. With this, $\frac{\partial x^i}{\partial x^j} = \delta^i_j$ is rather intuitive.
(By the way, you might argue that the notation $\frac{\partial}{\partial x^j}$ is slightly imprecise: After all, it does depend on all coordinates, but the notation only mentions $x^j$.)