While reading Jacobi one should bear in mind, that during his time the $f$ in $f(x)$ was not yet officially called the function. This modern use only started after 1900 and after Frege, Dedekind, Peano and Cantor. (Fore more on this see Who first considered the $f$ in $f(x)$ as an object in itself, and who decided to call it a function?.) Prior to 1900 what was called the function was the $f(x)$ or the $y$ in $y=f(x)$, in other words the variable quantity, and not the "rule" $f$. This is also how Jacobi uses the word function below. Beware hence, that what he denotes with $f$ and calls a function, is actually a variable quantity. (In modern terminology: his $f$ is an object of type $\mathbb{R}$ and not of type $\mathbb{R}\to\mathbb{R}$.)

What follows is my translation of the german translation Ueber die Functionaldeterminante, 1941 p.2 ff:

Before I come to the actual subject matter, I will start with some remarks concerning the notation of partial differentials. And since this treatise will contain repeated talk about functions, which may or may not depend from one another, it seems appropriate to also add some elementary considerations about these.

$$2. $$

To distinguish the partial differentials from the ordinary ones, hence from those where all variable quantities are seen as functions of a single variabel, Euler and others put the partial differentials in between brackets. But since an accumulation of brackets becomes rather annoying for reading and writing, I have preferred to use the characteristic $$ d $$ to denote ordinary differentials and the characteristic $$ \partial $$ for partial differentials. Adopting this convention rules out misunderstandings. So if $f$ is a function of $x$ and $y$, I will write
$$ df =\frac{\partial f}{\partial x}dx +\frac{\partial f}{\partial y}dy. $$

Whenever a function contains only a single variabel, one may use the characteristic $d$ or $\partial$ indifferently. [...]

In order for the partial differentials, of a function which depends on more than one variabel, to be completely determined, it does not suffice to provide the function to be differentiated and the variable with respect to which to differentiate, one must moreover express which quantities remain constant during the differentiation. For suppose $f$ is a function of $x,x_{1},\ldots, x_{n}$. Take $n$ arbitrary functions $\omega_{1},\omega_{2},\ldots ,\omega_{n}$ of these variables and consider $f$ as a function of the variables $x,\omega_{1},\omega_{2},\ldots ,\omega_{n}$. Then, when $x_{1},\ldots, x_{n}$ remain constant, the $\omega_{1},\omega_{2},\ldots ,\omega_{n}$ will not longer be constants with changing $x$, and neither will, when $\omega_{1},\omega_{2},\ldots ,\omega_{n}$ remain constant, the $x_{1},\ldots, x_{n}$ stay constant. The expression $\frac{\partial f}{\partial x}$ will hence assume completely different values, depending on whether these or those quantities are assumed constant during differentiation.

(In modern terminology one could rephrase this as follows: the vector field $\frac{\partial}{\partial x}$ associated to a coordinate chart $x,x_1,\ldots,x_n$ on a manifold, does not depend on the coordinate function $x$ alone, but also on the remaining coordinate functions $x_1,\ldots,x_n$.)

Suppose for example we introduce for a function $f$ of the two variables $x$ and $y$, an arbitrary function $u$ of $x$ and $y$, as the second independent variabel instead of $y$. Then the differential that was previously denoted with
$$ \frac{\partial f}{\partial x} $$ will now be expressed as $$ \frac{\partial f}{\partial x} + \frac{\partial f}{\partial u}\cdot\frac{ \partial{u}}{{\partial x}}, $$ so that the same signs $\frac{\partial f}{\partial x}$ denote completely different values, depending on whether $y$ or $u$ are kept constant, while differentiating $f$ with respect to $x$.

I will therefore in this treaties, whenever partial differentials are needed, not only express with the statement: $f$ is a function of $x,x_{1},\ldots, x_{n}$, that $f$ depends on these variables, hence that it remains constant when these are constant, and changes, when they change — that would equally be valid, if instead of $x,x_{1},\ldots, x_{n}$ any other variables $\omega,\omega_{1},\ldots, \omega_{n}$ functions of these, were introduced as independent variables— rather when I say, $f$ is a function of the $x,x_{1},\ldots, x_{n}$ I want the following to be understood: whenever this function is partially differentiated, the differentiation should occurr in such a way, that of these variables only one changes while the others remain constant.

Further, if the formulas are to be free of ambiguities, the notation should not only indicate the variable with respect to which the differentiation is occurring, but also the whole system of independent variables, the function of which is being differentiated, so that one may recognise which quantities remain constant during differentiation. And this is all the more necessary, since it is unavoidable that in the same calculation or even in one and the same formula, there appear partial differentials that refer to different systems of independent variables, for example in the above expression $$ \frac{\partial f}{\partial x} + \frac{\partial f}{\partial u}\cdot \frac{\partial{u}}{{\partial x}} $$ in which $f$ is seen as a function of $x$ and $u$, while $u$ is seen as a function of $x$ and $y$; this expression was precisely the one $\frac{\partial f}{\partial x}$ transitions into, wenn $u$ is introduced as independent variable instead of $y$. But if we write next to the dependent variable all the independent variables to which the partial differentiation refers to, then this expression can be depicted by the following formula, which is free of any ambiguity: $$ \frac{\partial f(x,y)}{\partial x} = \frac{\partial f(x,u)}{\partial x} + \frac{\partial f(x,u)}{\partial u}\cdot \frac{\partial{u(x,y)}}{{\partial x}}. $$

Certainly this notation, as well as every other imaginable notation that would allow one to completely determine any partial differentiation from the symbols alone, would become very cumbersome in more general investigations or more involved formulas, yes even impracticable, since with higher numbers of independent variables and more terms it might happen that a formula, which can be expressed in a single line, takes up a whole page. Certainly one should place the highest value on a notation that eliminates any ambiguity, and which makes any formula understandable on its own, without oral clarifications. But when it was possible without to much disadvantage, and in view of the enormous and unavoidable verbosity of the notation, I settled for the shorter notation of differentials, that dispenses with the specification of the independent variables.

Let $f,f_1,f_2$ etc. be mutually independent functions of the variables $x,x_1,\ldots,x_n$, and let $x$ be a variable contained in $f$ [...] it follows that the $m+1$ equations \begin{align} f(x,x_1,\ldots,x_n)&=\omega\\ f_1(x,x_1,\ldots,x_n)&=\omega_1\\ &\ldots\\ f_m(x,x_1,\ldots,x_n)&=\omega_m \end{align} do not determine more than $m+1$ quantities.

Here these $f$ are again variables quantities of type $\mathbb{R}$ and not of type $\mathbb{R}\to \mathbb{R}$, and there is no need to write $(x,x_1,\ldots,x_n)$ next to the $f$'s in the equations.

While reading Jacobi one should bear in mind, that during his time the $f$ in $f(x)$ was not yet officially called the function. This modern use only started after 1900 and after Frege, Dedekind, Peano and Cantor. (For more on this see Who first considered the $f$ in $f(x)$ as an object in itself, and who decided to call it a function?.) Prior to 1900 what was called the function was the $f(x)$ or the $y$ in $y=f(x)$, in other words the variable quantity, and not the "rule" $f$. This is also how Jacobi uses the word function below. Beware hence, that what he denotes with $f$ and calls a function, is actually a variable quantity. (In modern terminology: his $f$ is an object of type $\mathbb{R}$ and not of type $\mathbb{R}\to\mathbb{R}$.)

What follows is my translation of the German translation Ueber die Functionaldeterminante, 1941 p.2 ff:

Before I come to the actual subject matter, I will start with some remarks concerning the notation of partial differentials. And since this treatise will contain repeated talk about functions, which may or may not depend from one another, it seems appropriate to also add some elementary considerations about these.

$$2. $$

To distinguish the partial differentials from the ordinary ones, hence from those where all variable quantities are seen as functions of a single variabel, Euler and others put the partial differentials in between brackets. But since an accumulation of brackets becomes rather annoying for reading and writing, I have preferred to use the characteristic $$ d $$ to denote ordinary differentials and the characteristic $$ \partial $$ for partial differentials. Adopting this convention rules out misunderstandings. So if $f$ is a function of $x$ and $y$, I will write
$$ df =\frac{\partial f}{\partial x}dx +\frac{\partial f}{\partial y}dy. $$

Whenever a function contains only a single variable, one may use the characteristic $d$ or $\partial$ indifferently. [...]

In order for the partial differentials, of a function which depends on more than one variable, to be completely determined, it does not suffice to provide the function to be differentiated and the variable with respect to which to differentiate; one must moreover express which quantities remain constant during the differentiation. For suppose $f$ is a function of $x,x_{1},\dotsc, x_{n}$. Take $n$ arbitrary functions $\omega_{1}, \omega_{2}, \dotsc, \omega_{n}$ of these variables and consider $f$ as a function of the variables $x,\omega_{1}, \omega_{2}, \dotsc, \omega_{n}$. Then, when $x_{1}, \dotsc, x_{n}$ remain constant, the $\omega_{1}, \omega_{2}, \dotsc, \omega_{n}$ will no longer be constant with changing $x$, and neither will, when $\omega_{1}, \omega_{2}, \dotsc, \omega_{n}$ remain constant, the $x_{1}, \dotsc, x_{n}$ stay constant. The expression $\frac{\partial f}{\partial x}$ will hence assume completely different values, depending on whether these or those quantities are assumed constant during differentiation.

(In modern terminology one could rephrase this as follows: the vector field $\frac{\partial}{\partial x}$ associated to a coordinate chart $x,x_1,\ldots,x_n$ on a manifold, does not depend on the coordinate function $x$ alone, but also on the remaining coordinate functions $x_1, \dotsc, x_n$.)

Suppose for example we introduce for a function $f$ of the two variables $x$ and $y$, an arbitrary function $u$ of $x$ and $y$, as the second independent variable instead of $y$. Then the differential that was previously denoted by $$ \frac{\partial f}{\partial x} $$ will now be expressed as $$ \frac{\partial f}{\partial x} + \frac{\partial f}{\partial u}\cdot\frac{ \partial{u}}{{\partial x}}, $$ so that the same signs $\frac{\partial f}{\partial x}$ denote completely different values, depending on whether $y$ or $u$ are kept constant, while differentiating $f$ with respect to $x$.

I will therefore in this treatise, whenever partial differentials are needed, not only express with the statement: $f$ is a function of $x, x_{1}, \dotsc, x_{n}$, that $f$ depends on these variables, hence that it remains constant when these are constant, and changes, when they change — that would equally be valid, if instead of $x, x_{1}, \dotsc, x_{n}$ any other variables $\omega, \omega_{1}, \dotsc, \omega_{n}$ functions of these, were introduced as independent variables — rather when I say, $f$ is a function of the $x, x_{1} , \dotsc, x_{n}$ I want the following to be understood: whenever this function is partially differentiated, the differentiation should occur in such a way, that of these variables only one changes while the others remain constant.

Further, if the formulas are to be free of ambiguities, the notation should not only indicate the variable with respect to which the differentiation is occurring, but also the whole system of independent variables, the function of which is being differentiated, so that one may recognise which quantities remain constant during differentiation. And this is all the more necessary, since it is unavoidable that in the same calculation or even in one and the same formula, there appear partial differentials that refer to different systems of independent variables, for example in the above expression $$ \frac{\partial f}{\partial x} + \frac{\partial f}{\partial u}\cdot \frac{\partial{u}}{{\partial x}} $$ in which $f$ is seen as a function of $x$ and $u$, while $u$ is seen as a function of $x$ and $y$; this expression was precisely the one $\frac{\partial f}{\partial x}$ transitions into, when $u$ is introduced as independent variable instead of $y$. But if we write next to the dependent variable all the independent variables to which the partial differentiation refers, then this expression can be depicted by the following formula, which is free of any ambiguity: $$ \frac{\partial f(x,y)}{\partial x} = \frac{\partial f(x,u)}{\partial x} + \frac{\partial f(x,u)}{\partial u}\cdot \frac{\partial{u(x,y)}}{{\partial x}}. $$

Certainly this notation, as well as every other imaginable notation that would allow one to completely determine any partial differentiation from the symbols alone, would become very cumbersome in more general investigations or more involved formulas, yes even impracticable, since with higher numbers of independent variables and more terms it might happen that a formula, which can be expressed in a single line, takes up a whole page. Certainly one should place the highest value on a notation that eliminates any ambiguity, and which makes any formula understandable on its own, without oral clarifications. But when it was possible without too much disadvantage, and in view of the enormous and unavoidable verbosity of the notation, I settled for the shorter notation of differentials, that dispenses with the specification of the independent variables.

Let $f, f_1, f_2$ etc. be mutually independent functions of the variables $x, x_1, \ldots, x_n$, and let $x$ be a variable contained in $f$ [...] it follows that the $m+1$ equations \begin{align} f(x,x_1,\ldots,x_n)&=\omega\\ f_1(x,x_1,\ldots,x_n)&=\omega_1\\ &\ldots\\ f_m(x,x_1,\ldots,x_n)&=\omega_m \end{align} do not determine more than $m+1$ quantities.

Here these $f$ are again variables quantities of type $\mathbb{R}$ and not of type $\mathbb{R}\to \mathbb{R}$, and there is no need to write $(x, x_1, \dotsc, x_n)$ next to the $f$'s in the equations.

Suppose for example we introduce for a function $f$ of the two variables $x$ and $y$, an arbitrary function $u$ of $x$ and $y$, as the second independent variabel instead of $y$. Then the differential that was previously denoted with
$$ \frac{\partial f}{\partial x} $$ will now be expressed as $$ \frac{\partial f}{\partial x} + \frac{\partial f}{\partial u}\cdot\frac{ \partial{u}}{{\partial x}}, $$ so that the same signs $\frac{\partial f}{\partial x}$ denote completely different values, depending on whether $y$ or $u$ are kept constant, while differentiating $f$ with respect to $x$.

I will therefore in this treaties, whenever partial differentials are needed, not only express with the statement: $f$ is a function of $x,x_{1},\ldots, x_{n}$, that $f$ depends on these variables, hence that it remains constant when these are constant, and changes, when they change — that would equally be valid, if instead of $x,x_{1},\ldots, x_{n}$ any other variables $\omega,\omega_{1},\ldots, \omega_{n}$ functions of these, were introduced as independent variables— rather when I say, $f$ is a function of the $x,x_{1},\ldots, x_{n}$ I want the following to be understood: whenever this function is partially differentiated, the differentiation should occurr in such a way, that of these variables only one changes while the others remain constant.

Further, if the formulas are to be free of ambiguities, the notation should not only indicate the variable with respect to which the differentiation is occurring, but also the whole system of independent variables, the function of which is being differentiated, so that one may recognise which quantities remain constant during differentiation. And this is all the more necessary, since it is unavoidable that in the same calculation or even in one and the same formula, there appear partial differentials that refer to different systems of independent variables, for example in the above expression $$ \frac{\partial f}{\partial x} + \frac{\partial f}{\partial u}\cdot \frac{\partial{u}}{{\partial x}} $$ in which $f$ is seen as a function of $x$ and $u$, while $u$ is seen as a function of $x$ and $y$; this expression was precisely the one $\frac{\partial f}{\partial x}$ transitions into, wenn $u$ is introduced as independent variable instead of $y$. But if we write next to the dependent variable all the independent variables to which the partial differentiation refers to, then this expression can be depicted by the following formula, which is free of any ambiguity: $$ \frac{\partial f(x,y)}{\partial x} + \frac{\partial f(x,u)}{\partial u}\cdot \frac{\partial{u(x,y)}}{{\partial x}}. $$

Suppose for example we introduce for a function $f$ of the two variables $x$ and $y$, an arbitrary function $u$ of $x$ and $y$, as the second independent variabel instead of $y$. Then the differential that was previously denoted with
$$ \frac{\partial f}{\partial x} $$ will now be expressed as $$ \frac{\partial f}{\partial x} + \frac{\partial f}{\partial u}\cdot\frac{ \partial{u}}{{\partial x}}, $$ so that the same signs $\frac{\partial f}{\partial x}$ denote completely different values, depending on whether $y$ or $u$ are kept constant, while differentiating $f$ with respect to $x$.

I will therefore in this treaties, whenever partial differentials are needed, not only express with the statement: $f$ is a function of $x,x_{1},\ldots, x_{n}$, that $f$ depends on these variables, hence that it remains constant when these are constant, and changes, when they change — that would equally be valid, if instead of $x,x_{1},\ldots, x_{n}$ any other variables $\omega,\omega_{1},\ldots, \omega_{n}$ functions of these, were introduced as independent variables— rather when I say, $f$ is a function of the $x,x_{1},\ldots, x_{n}$ I want the following to be understood: whenever this function is partially differentiated, the differentiation should occurr in such a way, that of these variables only one changes while the others remain constant.

Further, if the formulas are to be free of ambiguities, the notation should not only indicate the variable with respect to which the differentiation is occurring, but also the whole system of independent variables, the function of which is being differentiated, so that one may recognise which quantities remain constant during differentiation. And this is all the more necessary, since it is unavoidable that in the same calculation or even in one and the same formula, there appear partial differentials that refer to different systems of independent variables, for example in the above expression $$ \frac{\partial f}{\partial x} + \frac{\partial f}{\partial u}\cdot \frac{\partial{u}}{{\partial x}} $$ in which $f$ is seen as a function of $x$ and $u$, while $u$ is seen as a function of $x$ and $y$; this expression was precisely the one $\frac{\partial f}{\partial x}$ transitions into, wenn $u$ is introduced as independent variable instead of $y$. But if we write next to the dependent variable all the independent variables to which the partial differentiation refers to, then this expression can be depicted by the following formula, which is free of any ambiguity: $$ \frac{\partial f(x,y)}{\partial x} = \frac{\partial f(x,u)}{\partial x} + \frac{\partial f(x,u)}{\partial u}\cdot \frac{\partial{u(x,y)}}{{\partial x}}. $$