I Relation between the momentum operator and the Hamiltonian gradient operator

Summary
quantum physics in solids - momentum operator
Is there a relationship between the momentum operator matrix elements and the following:

<φ|dH/dkx|ψ>

where kx is the Bloch wave number

such that if I have the latter calculated for the x direction as a matrix, I can get the momentum operator matrix elements from it?
 

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