Sometimes I see the multivariable chain rule defined solely in terms of partial derivatives $$ \frac{df}{dt} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial t}. $$ And sometimes I see it defined using a mix of partial derivatives and total derivatives $$ \frac{df}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}. $$ And sometimes I see it defined using a mix of partial derivatives and differentials, which I guess is equivalent to the 2nd definition $$ df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y} dy. $$ Needless to say, this is all quite confusing for someone moving beyond elementary calculus but not in a formal math program.
Can anyone enlighten me as to whether these different expressions mean the same thing or different things? Are authors being imprecise? What are good pedagogical references illustrating the meanings behind these different definitions?
The second notation is preferable if $x$ and $y$ (and therefore $f$, in the end) depend only on a single variable $t$, so that the derivatives $dx/dt$ and $dy/dt$ are just ordinary single-variable derivatives. (Not total derivatives, that's something slightly different.)
But an ordinary derivative $dx/dt$ can be written as a partial derivative $\partial x/\partial t$ without much risk of confusion, and if you do that, you get the first notation.
The first notation is also what you would use in case $x$ and $y$ depend on other variables besides $t$, except that then you must also write a partial derivative $\partial f/\partial t$ on the left-hand side.