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 $t$.
