Preamble

$Assumptions=({x,b,σ,left,right}∈ Reals)&&({σ,right}>0)&&({x,left}>=0) && left < right && b>=1

Define the link function between covariates and scale parameter

σ(z_,β_):=Exp(zβ)

Half-normal

g[x_,σ_] := Exp[-x^2/2*σ^2]
f[x_,σ_,left_,right_] := g[x,σ]/Integrate[g[x,σ],{x,left,right}]

D[f[x,σ,left, right],σ]

Simplify[D[f[x,σ,left, right],σ]]

Simplify[D[Log[f[x,σ,left, right]],σ]]

Hazard-rate

g[x_,σ_,b_]:=1-Exp[-(x/σ)^(-b)]
f[x_,σ_,b_, left_,right_] := g[x,σ,b]/Integrate[g[x,σ,b],{x,left,right}]

Simplify[D[f[x,σ,b,0, right],σ],σ in Reals]