derivative of multivariate equation with nested sum

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Hello,

I often have to deal with functions like the one below, take derivatives and so on. I would really like to know if I could use a CAS like SAGE to do this tedious and error prone calculations but I couldn't find a similar kind of function in the docs and tutorials.

My questions are:

• how can I write this function in SAGE ?

for $x\in \mathbf{R}^p; v \in \mathbf{R}^{p \times k}$
$$y(x, v) := \sum^p_{i=1} \sum^p_{j>i} \sum_{f=1}^k v_{i,f} v_{j,f} x_i x_j = \sum^p_{i=1} \sum^p_{j>i} \langle v_{:,i}, v_{;,j} \rangle x_i x_j$$

• calculate the partial derivatives $\frac{\partial y(x,v)}{\partial v_{i,j}}$ ?
• or the the derivative with respect to the column-vector $\frac{\partial y(x,v)}{\partial v_{:, i} }$ ?

Or is there a better way to work with this kind of function in SAGE? (the function above is only an example)

Thanks