Notational confusion of multivariable derivatives

I think there's several different confusions that arise from multivariable derivative notation:

• The thing where $\frac{\partial w}{\partial t}$ can mean two different things on LHS and RHS when $t$ is used as both an initial and intermediate variable. (See Folland for details.)
• The thing where if $f(x,y)=(x^2,y^2)$ then $\frac{\partial f}{\partial x}(x,x)$ feels like it might be $(2x,2x)$ even though it's actually $(2x,0)$. (Example from Tao.) See also [1]
• Total derivative vs derivative matrix

The derivative as a linear transformation in the several variable case and a number in the single-variable case

• The thing where the total derivative for $n=m=1$ "should" be a function but people treat it as a number. Refer to "Appendix A: Perorations of Dieudonne" (p. 337) in Pugh's Real Mathematical Analysis.

See also

• Summary table of multivariable derivatives