The definition for "total derivative" given at wikipedia here gives the formula for $f(t,x(t),y(t))$ as: $$\frac{d}{dt}\big(f(t,x(t),y(t)\big)=\frac{\partial f}{\partial t} \frac{dt}{dt}+ \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}$$
why does this formula not use only partial derivatives? (i.e. what is the point of the $\frac{dx}{dt}$ term (and similar for y and t)?
Alternatively, in this post with a proof of this total derivative formula, why is it that $$D_a(g) = \displaystyle \begin{pmatrix}dx/dt\\dy/dt\end{pmatrix}$$? since $D_a(g)$ is the jacobian, which should be a matrix of partial derivatives?
My guess is that it is because, since $x$ and $y$ are only a function of $t$, the total and partial derivatives are the same thing. but in general where $x$ and $y$ are not only a function of $t$, this logic does not hold
Thanks
Yes, your guess is basically correct - when $x$ is a function only of $t$, $\frac{dx}{dt}$ and $\frac{\partial x}{\partial t}$ are the same, basically by definition.
To elaborate a little more, though: It wouldn't be incorrect to use $\frac{\partial x}{\partial t}$ here, just misleading. The notation $\frac{\partial x}{\partial t}$ says "the derivative of $x$ with respect to $t$", but it heavily implies "...where $x$ is also a function of other variables". Similarly, if I said "some bachelors are unmarried", I would be saying something that is technically true, but suggesting the additional comment "some bachelors are not".