# Differentiation of a vector

I have a matrix $$A$$ which is $$n$$ by $$m$$ and a vector $$b$$ which is $$m$$ by 1. I have the following expression:

$$\frac{\delta}{\delta b} \mathbf{1}^{T}f(Ab)$$

Where $$\mathbf{1}$$ is a vector of ones which is $$n$$ by $$1$$, $$f$$ is a function which is applied elementwise and $$^{T}$$ represents transpose.

How can I express this just in terms of $$f'$$, $$A$$ and $$b$$? And does anyone have any resources on vector differentiation that could help me.

Let $$y=Ab,\,$$ then the differential of the elementwise function $$f(y)$$ is given by \eqalign{ df &= f'\odot dy \cr } where $$\odot$$ is the elementwise/Hadamard product.
Now find the differential and gradient of the cost function. \eqalign{ \phi &= 1:f \cr d\phi &= 1:df = 1:(f'\odot dy) = f':dy = f':A\,db = A^Tf':db \cr \frac{\partial\phi}{\partial b} &= A^Tf' \cr } where : is the trace/Frobenius product, i.e. $$\,A:B={\rm Tr}(A^TB)$$