phi1[t_]:=Tanh[Pi/2 Sinh[t]]

Plot[phi1[t],{t,-3,3}]

Plot[phi1'[t],{t,-3,3}]

f[x_]:=1/(x+2)

Plot[f[x],{x,-1,1}]

F[t_]:=f[phi1[t]]phi1'[t]

Plot[F[t],{t,-5,5},PlotRange->All]

f1[x_]:=1/Sqrt[1-x^2]

Plot[f1[x],{x,-1,1}]

F1[t_]:=f1[phi1[t]]phi1'[t]

Plot[F1[t],{t,-3,3},PlotRange->All]

phi2[t_]:=Sinh[Pi/2 Sinh[t]]

Plot[phi2[t],{t,-2,2}]

Plot[phi2'[t],{t,-2,2}]

g[x_]:=1/(x^2+1)

Plot[g[x],{x,-5,5}]

G[t_]:=g[phi2[t]]phi2'[t]

Plot[G[t],{t,-5,5},PlotRange->All]

phi3[t_]:=Exp[t-Exp[-t]]

Plot[phi3[t],{t,-5,5}]

Plot[phi3'[t],{t,-5,5}]

h[x_]:=x^5 Exp[-x]

Plot[h[x],{x,0,10}]

H[t_]:=h[phi3[t]]phi2'[t]
Plot[H[t],{t,-5,5},PlotRange->All]