Symbolic computation of gradient

[v0id]

I'm wondering if there are any convenient symbolic "shortcuts" (i.e. abuse of notation) that enable one to compute the gradient with respect to a certain vector, without decomposing the computation into the vector's individual elements and differentiating with respect to each element. For example:
$$\nabla_x \left( \frac{1}{|{\bf x}^{'} - {\bf x}|} \right) = \frac{{\bf x}^{'} - {\bf x}}{|{\bf x}^{'} - {\bf x}|^3}$$
Besides the obvious method of evaluating $$\frac{\partial}{\partial x_1}$$ and so on, is there a faster method of symbolic computation?

 

[ansrivas]

If you consider
$$\sqrt{x^Tx}=|x|$$
then symbolic ideas help to get to the final expression:

$$\nabla_x \left( (x'-x)^{T}(x'-x) \right)^{-1/2} = \left( (x'-x)^{T}(x'-x) \right)^{-3/2} (x'-x)$$

 

[v0id]

Yes, that works very well. Thanks a lot.