Basic question on notation of partial derivatives I've seen the (single variable) chain rule written in two ways:
$\left(f(g(x)) \right)'=f'(g(x))\cdot g'(x)$ and $\frac {dy}{dx}=\frac {dy}{dt}\cdot \frac {dt}{dx}$.
The second one is more commonly used later in multivariate calc.
My doubt is: When you use the Leibnitz notation, how do you know where each $\frac {\partial z}{\partial w}$ is evaluated at?
In the single variate chain rule, $\dfrac {dy}{dt}$ is evaluated at $t$, but this is nowhere in the formula.
 A: Since $y=y(t)=y(t(x))$ then the rule is
$$\frac{dy}{dx}(x_0)=\frac{dy}{dt}(t(x_0))\frac{dt}{dx}(x_0),$$
for functions in one variable. 
For functions on more than one variable
you need to specify them explicitly what you have in mind. 

As an example in several variables, let us consider 
$$F=F(x,y,z)=F(x(v,w),y(v,w),z(v,w)),$$
so, maybe you are interested in the calculation of $\frac{\partial F}{\partial v}$ and $\frac{\partial F}{\partial v}$, 
then
$$\frac{\partial F}{\partial v}=\frac{\partial F}{\partial x}\frac{\partial x}{\partial v}+\frac{\partial F}{\partial y}\frac{\partial y}{\partial v}+\frac{\partial F}{\partial z}\frac{\partial z}{\partial v},$$
and 
$$\frac{\partial F}{\partial w}=\frac{\partial F}{\partial x}\frac{\partial x}{\partial w}+\frac{\partial F}{\partial y}\frac{\partial y}{\partial w}+\frac{\partial F}{\partial z}\frac{\partial z}{\partial w},$$
and on evaluation at a point $p=(x(v_0,w_0),y(v_0,w_0,z(v_0,w_0))$ then it would be
$$\frac{\partial F}{\partial v}(v_0,w_0)=\frac{\partial F}{\partial x}(p)\frac{\partial x}{\partial v}(v_0,w_0)+\frac{\partial F}{\partial y}(p)\frac{\partial y}{\partial v}(v_0,w_0)+\frac{\partial F}{\partial z}(p)\frac{\partial z}{\partial v}(v_0,w_0).$$
Similarly  for $\frac{\partial F}{\partial w}(v_0,w_0)$.
