TITLE: Changing variables in a partial derivative
QUESTION [5 upvotes]: If I have the equation and function $$f_1(x_1,x_2,x_3,...,x_n) = 0,\qquad x_1 = g_1(y_1, y_2, y_3,...,y_m)$$ then what is $\frac {\partial f_1}{\partial x_1}$ in terms of $g_1$ and $y_i$?

REPLY [4 votes]: Not sure if that is what you need, 
but if you have a function of $n$ variables $f_{1}(x_{1}, x_{2}, x_{3},...,x_{n})$, where 
$x_{1} = g_{1}(y_{1}, y_{2}, y_{3},...,y_{m})$, $x_{2} = g_{2}(y_{1}, y_{2}, y_{3},...,y_{m}),\dots, x_{n} = g_{n}(y_{1}, y_{2}, y_{3},...,y_{m})$  are also some functions of $m$ variables, then you can find partial derivatives of $f_{1}$ with respect to $y_{i},  i=\overline{1,n} $ using the formula:
$$\frac{\partial f_{1}}{\partial y_{i}} = \frac{\partial f_{1}}{\partial x_{1}}\frac{\partial x_{1}}{\partial y_{i}} + \frac{\partial f_{1}}{\partial x_{2}}\frac{\partial x_{2}}{\partial y_{i}} + \ldots + \frac{\partial f_{1}}{\partial x_{n}}\frac{\partial x_{n}}{\partial y_{i}} = \sum_{k=1}^{n} \frac{\partial f_{1}}{\partial x_{k}}\frac{\partial x_{k}}{\partial y_{i}}$$