Suppose I have the following Jacobian
$J_{ij}(\underline{\Delta\sigma})=\displaystyle\frac{\partial \Delta\varepsilon_i}{\partial \Delta\sigma_j} \in \mathbb{R}^{6\times6}$
within a Newton-Raphson algorithm. $\underline{\Delta\varepsilon}\in\mathbb{R}^{6\times1}$ and $\underline{\Delta\sigma}\in\mathbb{R}^{6\times1}$ are vector valued functions. Looking at the first component of the Jacobian, suppose that $\Delta\varepsilon_1$ has the following dependencies
$\Delta\varepsilon_1=\Delta\varepsilon_1(g(\Delta\sigma_1), \Delta\sigma_1),$
with some function $g$, so that we are dealing with a function composition. To my understanding, if I take the partial derivative $\frac{\partial \Delta\varepsilon_1}{\partial \Delta\sigma_1}$, I would need to keep $g(\Delta\sigma_1)$ constant. Thereby, I would neglect something in the rate of change, because when I change $\Delta\sigma_1$, also $g(\Delta\sigma_1)$ will change.
I could take the total derivative of the expression
$\displaystyle\frac{{\rm d} \Delta\varepsilon_1}{{\rm d} \Delta\sigma_1} = \displaystyle\frac{\partial \Delta\varepsilon_1}{\partial g} \frac{\partial g}{\partial \Delta\sigma_1} + \frac{\partial \Delta\varepsilon_1}{\partial \Delta\sigma_1}$
and use this in the Jacobian instead to consider the full rate of change at a given point, so that the definition of the Jacobian somewhat changes to
$J_{ij}(\underline{\Delta\sigma})=\displaystyle\frac{{\rm d} \Delta\varepsilon_i}{{\rm d} \Delta\sigma_j}.$
Would that work out or am I doing something fundamentally wrong?
TLDR: I take partial derivatives of function compositions (which are somewhat arbitrary) and feel that I should take the total derivative instead in the Newton-Raphson algorithm.
I believe you should take the total derivative. Whenever we define a partial derivative, we must (implicitly) choose what to keep constant. For the purposes of the Jacobian, $\frac{\partial \Delta\varepsilon_1}{\partial \Delta\sigma_1}$ means you are differentiating $\Delta\varepsilon_1$ w.r.t $\Delta\sigma_1$ while assuming that $\Delta\sigma_i$ for $i\neq 1$ are constant, because the purpose is to 'separate' the influence of $\Delta\sigma_1$ on $\Delta\varepsilon_1$ from the influence of the other $\Delta\sigma_i$.
So even though $\Delta\varepsilon_1$ may be written as a two-variable function of $g$ and $\Delta\sigma_1$, you should still consider the effect of $g$ since it itself is influenced by $\Delta\sigma_1$, which is the primary variable whose influence of you're studying by calculating $\frac{\partial \Delta\varepsilon_1}{\partial \Delta\sigma_1}$.