sol = DSolve[{n'[t]==-4 Pi/τ (1 - n[t])n[t],n[0]==n0},n[t],t]

p[t_, τ_, n0_]=D[n[t]/.sol[[1]],t]

Manipulate[Plot[p[t, τ, Tanh[n0]],{t,0,50},PlotRange->Full],{τ,0.01,100},{n0,0.3,4}]