Given a function $f(x, y)$ and knowing that both x and y are functions of $s$ and $t$, can we use the chain rule to find, for example, the partial derivative of f with respect to s ($\frac{\partial f}{\partial s}$).
To be more specific, the example could be: $$f(1,3) = 8;$$ $$\frac{\partial f}{\partial x} = 3;$$ $$\frac{\partial f}{\partial y} = -1;$$ $$\frac{\partial x}{\partial s} = 7;$$ $$\frac{\partial y}{\partial s} = 5;$$
And let's say that I have to find the partial of f with respect to s. $$\frac{\partial f}{\partial s} = \, ?$$
My reasoning is that this, in itself, is not enough. I think we need to be presented with the function itself so that we can look into its properties. I am mainly thinking about differentiability here. I seem to recal that the chain rule cannot be used if the function itself is not differentiable.
All help is much appreciated! Thank you in advance.