# Multivariable calculus question

evilcman
I have a scalar function f dependent on a few variables $x_i$, and I would like to change variables, so that $$y_i = \sum_j {M_{ij} x_j},$$ where M is an invertible matrix independent of the x_i-s, and compute:
$$\frac{\partial f}{\partial x_i} = \frac{\partial f}{\partial \left( \sum_j {M^{-1}_{ij} y_j} \right)} = \sum_j Q_{ij} \frac{\partial f}{\partial y_j}$$

I suggest that the last identity is true for some matrix Q. Is there a general formula for Q?

$$\frac{\partial f}{\partial x_i} = \sum_j \frac{\partial y_j}{\partial x_i} \frac{\partial f}{\partial y_j}= \sum_j Q_{ij} \frac{\partial f}{\partial y_j}$$
$$Q_{ij}=\frac{\partial y_j}{\partial x_i}$$