# Partial derivative of a composite function $\mathbb{R}^n \to \mathbb{R}^n$

I am trying to understand a proof but I am stuck on this technical bit:

Apart from the small typo highlighted, I don't really see how to get the big formula for the partial derivative of $v_i$

What I keep getting is the following:

$$\partial_i v_i(x)=\displaystyle \sum_{j,k=1}^{n} d_{i,j}(x) \partial_k (w_j(f(x))\partial_i( f_k) +\sum_{j=1}^n w_j(f(x)\partial_i d_{i,j}$$

where $\partial_k (w_j(f(x))=\displaystyle \frac{\partial w_j(f(x))}{\partial f_k}$

Later in the book an expression very similar to the one I get appears...are they the same thing? Is there a huge typo?

EDIT: I am thinking it could be that $W\not = w$ for some reason, is it a standard notation for something I don't know? I don't really see why they would have changed to capital letters otherwise...

Thank you very much!!

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Did you also apply $\partial_i$ to the $d_{i,j}(x)$-Term? – Quickbeam2k1 Mar 4 '13 at 9:31
@Quickbeam2k1 ah yeah you're right I forgot the second part! – Moritzplatz Mar 4 '13 at 9:32
And in the first sum, it should be $d_{i,j}(x)$ and not $d_{j,k}$, and it should be put at the end in order not to confuse the reader. – user10676 Mar 4 '13 at 9:37
@user10676 are you talking about the bit I copied from the book? I think it's wrong as well.. – Moritzplatz Mar 4 '13 at 9:38
For me too, the book is wrong. I may be misunderstanding the notations. However the typo is very bad (writing $divv(x)$ for $\operatorname{div} v(x)$...). – user10676 Mar 4 '13 at 9:42