Partial derivatives vs. Total Derivatives for chain rule. If I had a function $f(x,y)$ where $x=x(s,t)$ and $y=y(s,t)$ then 

$$ \frac{df}{dt} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial t}. $$

I have been reading this and have spotted in case 2 that 

$$ \frac{\partial f}{\partial t} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial t}. $$

$f$ has no explicit $t$ dependence so how can this be correct? How can I have a chain rule for partial derivatives? A chain rule implies that there is no explicit dependence on the variable that we are differentiating with respect to, and I thought partial derivatives only deal with explicit dependencies and not implicit? Thanks
In case $2$, it's not $\frac{\partial f}{\partial t}$, it's $\frac{\partial z}{\partial t}$, and $z$ is not independent of $t$.
An ordinary derivative--such as that of $f$ with respect to $t$, in your example is only valid if all the dependencies of $f$, all the dependencies of those dependencies, all the dependencies of those dependencies, and so on, can be shown to have a single common dependency $t$, explicit or otherwise. In all other cases, only a partial derivative applies, since $f$ then only depends partially on $t$. 

That is why a partial derivative is used in Case 2; $t$ is not a common dependency of the system $(f,x,y,s,t)$, as $s$ is in no way dependent upon $t$.