The boundaries for the solution of the linear Volterra integral equations of the second type of the form[image omitted] with unit source term and positive monotonically increasing convolution kernel were obtained as |f(t)|1, |f(t)|2 and |f(t)|4 in [R. Ling, Integral equations of Volterra type, J. Math. Anal. Appl. 64 (1978), pp. 381-397, R. Ling, Solutions of singular integral equations, Internat. J. Math. Math. Sci. 5 (1982), pp. 123-131.]. The sufficient conditions which are useful for finding the boundary such as |f(t)|2n of the solution of this equation were given, where 0t and n is a natural number, [I. Ozdemir and O. F. Temizer, The boundaries of the solutions of the linear Volterra integral equations with convolution kernel, Math. Comp. 75 (2006), pp. 1175-1199.]. In this paper, a method which ensures finding the boundaries of the derivative functions f', f'', ..., f(n+2) for n of the solution of the same equation has been developed.