# Differential derivatives of dependent variables

In my problem, I have six variables $X_1$, $X_2$, $X_3$, $Y_1$, $Y_2$, and $Y_3$ that are related using the following equations ($C$ is constant):

$$X_1=Y_1-C Y_2 cos(Y_3)$$ $$X_2=Y_1-C Y_2 cos(Y_3-2\pi/3)$$ $$X_3=Y_1-C Y_2 cos(Y_3-4\pi/3)$$

Therefore, we can derive variables $Y_1$, $Y_2$, and $Y_3$ as functions of variables $X_1$, $X_2$, and $X_3$ as:

$$Y_1=f(X_1, X_2, X_3)$$ $$Y_2=g(X_1, X_2, X_3)$$ $$Y_3=h(X_1, X_2, X_3)$$

Now my two questions are:

1. How can I derive $\frac{dY_2}{dX_1}$?
2. Is $\frac{dY_2}{dX_1}$ equal to $\frac{dX_1}{dY_2}$?

The partial derivatives of $X_1, X_2, X_3$ functions of $Y_1, Y_2, Y_3$ are easy to obtain.
The partial derivatives of $Y_1, Y_2, Y_3$ functions of $X_1, X_2, X_3$ require the inversion of a matrix (or solving the system of 3 linear equations). This is a boring task. Courage and good luck !