Partial Derivatives and defining a function. Define 


  f(x,y) = x+2y and w = x+y.  


What is $\frac{df}{dw}$? Does it make sense to define the partial derivative of a function f with respect to an arbitrary function w(not just x or y)? If so, what does this definition give for the specific example $f(x,y) = x+2y$ and $w = x+y$?
No, it's not meaningful unless you specify a coordinate system $(w,z)$ where $w$ is one of the coordinates. When computing $\partial f/\partial w$, you should first express $f$ in terms of $w$ and $z$, and then hold the other variable $z$ constant when you differentiate, so the result depends on what $z$ is.

Sometimes the notation
$$\left. \frac{\partial f}{\partial w} \right|_z$$
is used, to emphasize this.
$\dfrac{df}{dw} = \dfrac{df}{dx}\cdot \dfrac{dx}{dw} + \dfrac{df}{dy}\cdot \dfrac{dy}{dw}= \dfrac{\dfrac{df}{dx}}{\dfrac{dw}{dx}}+\dfrac{\dfrac{df}{dy}}{\dfrac{dw}{dy}}=\dfrac{1}{1}+\dfrac{2}{1}=3.$