F[x_]:=Max[0.001*x,Min[6.12+0.001*x,x]]
Plot[F[x],{x,-10,+20}]

D[F[x],x]

Plot[%,{x,-10,+20},PlotRange:>All]

G[x_]:= (6120/(1+Exp[-x+5]))

Plot[G[x],{x,-10,+20}]

6*G

D[G[x],x]

Plot[%,{x,-10,+20},PlotRange:>All]