This might be obvious, but I am not able to understand it. Would appreciate any help.
Let there be a function which can be decomposed into $\ k$ functions using addition $\ f = \sum_{i=1}^k f_i$. We want to find partial derivative of $\ f$ w.r.t. $\ y$. Is there something wrong in the following approach? If yes, then what is the problem. Is there a simple proof of the chain rule that contradicts this, or proves it? Thanks!
$$ \frac{\partial(f)}{\partial(y)} = \sum_{i=1}^k \frac{\partial(f_i)}{\partial(y)} = \sum_{i=1}^k \frac{\partial(f_i)}{\partial(x_i)}\cdot\frac{\partial(x_i)}{\partial(y)}$$