We say that the function f: [OMEGA] [right arrow] Y is H-differentiable (

differentiable in the sense of Hadamard) at the point a [member of] [OMEGA] if there exists u [member of] L(X,Y) such that

(2) the functions [[phi].sub.1[sigma]] and [[phi].sub.2[sigma]] are two

differentiable functions and [[[D.sup.1.sub.2][phi](x)].sup.[sigma]] = [[[phi]'.sub.2[sigma]](x), [[phi]'.sub.1[sigma]](x)], when [phi] is (2)-differentiable.

f is first-order

differentiable, so in convex domain D, first-order difference of f can be written in the following integral form:

Let J be a

differentiable application on [[OMEGA].sub.0].

Recently, Farid [3] extended Theorems 1 and 2 to functions of two variables that are

differentiable on their coordinates.

For nonnegative and

differentiable functions i, [m.sub.0], and [m.sub.1]: [[a.sub.0], [omega]] [right arrow] R, with [DELTA]m(a) = [m.sub.1](a)-[m.sub.0] (a) [greater than or equal to] 0 for all a [member of] [[a.sub.0], [omega]], the system of ODEs given by equations (1) and (2) with initial conditions S([a.sub.0]) = [S.sub.0] [greater than or equal to] 0, C([a.sub.0]) = [C.sub.0] [greater than or equal to] 0, and S([a.sub.0])+ C([a.sub.0]) > 0 has a unique solution S and C with N(a) = S(a) + C(a) [greater than or equal to] 0 for all a [member of] [[a.sub.0], [omega]].

If f (x) is

differentiable at x = a then it is continuous at x = a.

there exists a twice continuously

differentiable function v :

Let X be a real Banach space, and let [PHI], [PSI] : X [right arrow] R be two continuously Gateaux

differentiable functionals such that [inf.sub.X] [PHI] = [PHI](0) = [PSI](0) = 0.

As an application of the main results, it is shown that if we identify every fuzzy number with the corresponding equivalence class, there would be more

differentiable fuzzy functions than what is found in the literature.

Motivated by the above works, in this paper we give some applications and examples for trapezoidal and midpoint type inequalities when the intended function is

differentiable. Furthermore we consider integral quadrature formula and give an error estimate related to trapezoidal and midpoint formula.

The main disadvantage can be seen immediately by considering (7): the terminal condition [PSI] must be infinitely Malliavin

differentiable. In contradistinction, the viscosity solution given in [7] necessitates [PSI] to be only bounded and continuous.