How do I find $\frac{dy}{dx}$?

Question

I am

given $y = (x_1,..., x_N)$ and $F (y, x_1,..., x_N) = 0$.

I need to find $\frac{dy}{dx}$  in terms of the partials of $F$.

I have no idea at all after a lot of playing with it.

Answer 1

I guess, you mean $y = y(x_1,\dots,x_N)$. Then just differentiate $F$ w.r.t. $x_i$: $$ \frac{\partial}{\partial x_i}F(y,x) = F_y\cdot \frac{\partial y}{\partial x_i} + F_{x_i} $$ $$ \Downarrow $$ $$ \frac{\partial y}{\partial x_i} = -\frac{F_{x_i}}{F_y} $$ under the condition $F_y\neq 0$. More details are here.