In the Wikipedia page on Ricci calculus the following tensor derivative equation is given:
$$A_{\alpha \beta ..., \gamma}:= \frac \partial {\partial x^\gamma}A_{\alpha \beta ...}.$$
However, what does the term on the RHS mean? In particular, why are we taking the derivative of a tensor expression $A$ w.r.t. another tensor $x$ when that expression doesn't contain $x$? (Note, I'm only now starting to learn about tensors, though I have knowledge of vector calculus).
Look like the confusion is about $x^\gamma$: $x^\gamma$ is not a tensor, it's just a coordinates. As a tensor field, each component of $A$ is a function of $x^1, \cdots, x^n$, and
$$ \frac{\partial}{\partial x^\gamma} A_{\alpha \beta...}$$
is really just the partial derivative of the function $A_{\alpha\beta ...}$ with respect to $x^\gamma$. In general, $A_{\alpha\beta ... , \gamma}$ is not a tensor: this is explained in the section "covariant derivatives" in the same wiki page.