Total derivative notation help consider the function

$$f = f(x(t),y(t))$$

I know that its total derivative wrt t is

$$\frac {df}{dt} = \frac {\partial f} {\partial x} \frac {dx}{dt} + \frac {\partial f}{\partial y} \frac {dy}{dt}$$

and that the total derivative wrt x is
$$ \frac {df} {dx} = \frac {\partial f} {\partial x} \frac {dx}{dx} + \frac {\partial f} {\partial y} \frac {dy} {dx}$$

However I am not fully familiar with the notation and the forms in which it takes during more extreme conditions, such as the following, could anyone fill the blanks in for me?

1)$$f = f(x(t,w),y(t,w))$$
$$\frac {df}{dt} = ?$$
2)
$$f = f(x(g(t,w)),y(g(t,w)))$$
$$\frac {df}{dt} = ?$$
3)
$$f = f(x(g(t,w),z),y(g(t,w),z))$$
$$\frac {df}{dt} = ?$$
4)
$$f = f(x(g(t)),y(g(t)))$$
$$\frac {df}{dt} = ?$$

how would I go about writing these properly? Any answer will help illustrate the semantics in much greater detail
I shall write for the most extreme case (3)

$$\frac{df}{dt} =\frac{\partial f}{\partial x}\frac{\partial x}{\partial g}\frac{dg}{dt}+\frac{\partial f}{\partial x}\frac{\partial x}{\partial g}\frac{\partial g}{\partial w} \frac{dw}{dt}+\frac{\partial f}{\partial x}\frac{\partial x}{\partial z}\frac{dz}{dt}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial g}\frac{dg}{dt}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial g}\frac{\partial g}{\partial w} \frac{dw}{dt}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial z}\frac{dz}{dt}$$

A good rule of thumb could be imagining this as an expanding tree.
In general if you have the function $H(t)= F(G(t))$ and you want to find the derivative of $h$ with respect to $t$. You have the general rule: 
$$\frac{d H}{dt} = DF(G(t))(\frac{d G}{dt}).$$ It seems to be strange notation with respect to your level, but in this case you can take,for example, $$G(t)= (x(t),y(t))$$ and use $$Df((x,y))((z_1,z_2))= \frac{\partial f}{\partial x}z_1+ \frac{\partial f}{\partial y}z_2.$$