Which version of the total derivative formula is "more correct"?

Question

For $y=f(t,u_1,\ldots,u_m)$ with $u_i$ depending on $t$, i.e. $y$ not only directly but also indirectly depending on $t$:

Wolfram says: $\frac{\partial y}{\partial t}=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial u_1}\frac{\partial u_1}{\partial t}+ \ldots+\frac{\partial f}{\partial u_m}\frac{\partial u_m}{\partial t}$

while Wikipedia says:

$\frac{df}{dt}=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial u_1}\frac{du_1}{dt}+ \ldots+\frac{\partial f}{\partial u_m}\frac{du_m}{dt}$

Is it not unambiguous at Wolfram that partial and total derivative are used with the same symbols? From what I have read, total derivatives take into account the indirect effects of $t$, while partial derivatives do not. Even if the expression left to the equal sign here could be chosen freely (i.e. $\frac{df}{dt} \equiv \frac{\partial y}{\partial t}$): Does Wolfram not disregard that also $u_i$ could be some function that, again, depends indirectly on $t$? If I understand correctly, the partial derivative would neglect that, the total would not. Is this correct? Is Wolfram unprecise here because it uses $\frac{\partial u_1}{\partial t}$ instead of $\frac{du_1}{dt}$?

Also, the Wolfram expression for the total derivative confuses me... Using the partial derivative on a variable that is the result of a function seems even more unprecise. $\frac{\partial y}{\partial t}$ is the total derivative and $\frac{\partial f}{\partial t}$ is the partial derivative... what is the intuition/reasoning for this?

Answer 1

Notation for partial derivatives is often a bit sloppy and context-dependent, and requires a bit of interpretation on the part of the reader, and it's best to just learn to live with that. But I agree that it's clearer to use $d/dt$ for total derivative; how else can one tell that it's something other than $\partial/\partial t$?

If you want to be really precise, you'll have to write something like this: if from $f(t,u_1,\dots,u_n)$ we define the composite function $$ h(t) = f\bigl(t, g_1(t),\dots,g_n(t)\bigr) , $$ then $$ \frac{dh}{dt}(t) = \frac{\partial f}{\partial t}\bigl(t, g_1(t),\dots,g_n(t)\bigr) + \sum_{k=1}^n \frac{\partial f}{\partial u_k}\bigl(t, g_1(t),\dots,g_n(t)\bigr) \, \frac{dg_k}{dt}(t) . $$ (Under suitable differentiability assumptions, of course.)

Alternatively, using primes for ordinary derivatives and subscripts for partial derivatives: $$ h'(t) = f_t\bigl(t, g_1(t),\dots,g_n(t)\bigr) + \sum_{k=1}^n f_{u_k}\bigl(t, g_1(t),\dots,g_n(t)\bigr) \, g_k'(t) . $$