Product rule for Hadamard product differentation? Is there a "simple" solution to
$\bf \frac{\partial}{\partial w}\big(w \odot f(w)\big)$
assuming the matrix $\bf \frac{\partial f}{\partial w}$ is known? With simple I mean something like in the normal vector multiplication case 
$\bf \frac{\partial}{\partial w}\big(w^Tf(w)\big) = f(w) + \big[\frac{\partial f(w)}{\partial w}\big]^T w$
such that no other knowledge of $\bf f(w)$ is required.  

Let Diag denote the function which creates a diagonal matrix from a vector. 
Define some variables for convenience 
$$\eqalign{
 F &= \operatorname{Diag}(f) \cr
 W &= \operatorname{Diag}(w) \cr
 h &= w\odot f = f\odot w \cr
 J &= \frac{\partial f}{\partial w} \cr
\cr
}$$
Now find the differential and gradient of $h$
$$\eqalign{
 dh &= f\odot dw + w\odot df \cr
    &= F\,dw + W\,(J\,dw) \cr
\cr
\frac{\partial h}{\partial w} &= F + WJ \cr
}$$
