$F(\kappa, \nu)$ with $\kappa = \frac{x_1}{x_2}$ and $\nu = \frac{x_1}{x_3}$. Chain rule to $F_x$

Question

Let $\kappa = \frac{x_1}{x_2}$ and $\nu = \frac{x_1}{x_3}$. Consider a function $F(\kappa, \nu)$

How to apply chain rule to the following $$\frac{\partial F}{\partial x_1}$$

I am confused that since both $\kappa$ and $\nu$ are functions of $x_1$, how to use chain rule to derive it by chain rule? Can anyone please give me a hint?

Here $F$ for example is a polynomial!

Answer 1

By the chain rule,

$$ \dfrac{ \partial F }{\partial x_1} = \dfrac{ \partial F}{\partial \kappa} \frac{ \partial \kappa }{\partial x_1} + \dfrac{ \partial F}{\partial \nu} \frac{ \partial \nu }{\partial x_1} = \dfrac{ F_{\kappa } }{x_2} + \dfrac{ F_{\nu } }{x_3}$$

In general, if $F = F(f_1,...,f_n) $ and each $f_i = f_i(x_1,...,x_m)$, then the chain rule states that

$$ \dfrac{ \partial F }{\partial x_j} = \sum_{k=1}^n \dfrac{ \partial F}{\partial f_k }\dfrac{ \partial f_k }{\partial x_j}$$

where $j \in \{1,2,...,m \} $