Differentiation with respect to a vector function

Question

Suppose u is a smooth vector valued function say u : $ \mathbb{R}^n \rightarrow \mathbb{R}^m $ and $ W $ is a smooth scalar valued function, $W$ : $ \mathbb{R}^m \rightarrow \mathbb{R} $. What would $ W_u(u) $ be? I think that it could look like this $$ W_u(u) = [ \dfrac{\partial W\left( u\right) }{\partial u_{1}},\ldots , \dfrac{\partial W\left( u\right) }{\partial u_{m}} ] $$ but I feel that I'm missing something that has to do maybe with the coordinates $ (x_1, \ldots , x_n) $, I'm not sure. Sadly, I do not have a solid differential geometry background.