The concept of the fuzzy derivative was first introduced by Chang and Zadeh [14]; it was followed up by Dubois and Prade [15] who used the extension principle in their approach. Other methods have been discussed by Puri and Ralescu [4], Goetschel and Voxman [16], Kandel and Byatt [1, 2]. Lakshmikantham and Nieto introduced the concept of fuzzy differential equation in a metric space [17]. Puri and Ralescu in [4] introduced Hderivative (differentiability in the sense of Hukuhara) for fuzzy mappings and it is based on the difference of sets, as follows. Henceforth, we suppose for
Definition 3.1.
Let be a fuzzy function. One says, is differentiable at if there exists an element such that the limits
exist and are equal to Here the limits are taken in the metric space
The above definition is a straightforward generalization of the Hukuhara differentiability of a setvalued function. From [6, Proposition 4.2.8], it follows that Hukuhara differentiable function has increasing length of support. Note that this definition of derivative is very restrictive; for instance, in [9], the authors showed that if where is a fuzzy number and is a function with , then is not differentiable. To avoid this difficulty, the authors [9] introduced a more general definition of derivative for fuzzynumbervalued function. In this paper, we consider the following definition [11].
Definition 3.2.
Let and fix One says is (1)differentiable at , if there exists an element such that for all sufficiently near to , there exist and the limits (in the metric )
is (2)differentiable if for all sufficiently near to , there exist and the limits (in the metric )
If is differentiable at , we denote its first derivatives by , for
Example 3.3.
Let and define by for all . If is differentiable at , then is generalized differentiable on and we have . For instance, if , is (1)differentiable. If then is (2)differentiable.
Remark 3.4.
In the previous definition, (1)differentiability corresponds to the Hderivative introduced in [4], so this differentiability concept is a generalization of the Hderivative and obviously more general. For instance, in the previous example, for with we have .
Remark 3.5.
In [9], the authors consider four cases for derivatives. Here we only consider the two first cases of [9, Definition 5]. In the other cases, the derivative is trivial because it is reduced to crisp element (more precisely, . For details, see [9, Theorem 7]).
Theorem 3.6.
Let be fuzzy function, where for each .
(i)If is (1)differentiable, then and are differentiable functions and .

(ii)
If is (2)differentiable, then and are differentiable functions and .
Proof.
See [11].
Now we introduce definitions for higherorder derivatives based on the selection of derivative type in each step of differentiation. For the sake of convenience, we concentrate on the secondorder case.
For a given fuzzy function , we have two possibilities (Definition 3.2) to obtain the derivative of ot : and . Then for each of these two derivatives, we have again two possibilities: and respectively.
Definition 3.7.
Let and . One says say is differentiable at , if exists on a neighborhood of as a fuzzy function and it is differentiable at . The second derivatives of are denoted by for .
Remark 3.8.
This definition is consistent. For example, if is and differentiable simultaneously at , then is (1) and (2)differentiable around . By remark in [9], is a crisp function in a neighborhood of .
Theorem 3.9.
Let or be fuzzy functions, where .
(i)If is (1)differentiable, then and are differentiable functions and .

(ii)
If is (2)differentiable, then and are differentiable functions and .

(iii)
If is (1)differentiable, then and are differentiable functions and .

(iv)
If is (2)differentiable, then and are differentiable functions and .
Proof.
We present the details only for the case (i), since the other cases are analogous.
If and , we have
and multiplying by we have
Similarly, we obtain
Passing to the limit, we have
This completes the proof of the theorem.
Let be a positive integer number, pursuing the abovecited idea, we write to denote the thderivatives of at with for . Now we intend to compute the higher derivatives (in generalized differentiability sense) of the difference of two fuzzy functions and the product of a crisp and a fuzzy function.
Lemma 3.10.
If are thorder generalized differentiable at in the same case of differentiability, then is generalized differentiable of order at and . (The sum of two functions is defined pointwise.)
Proof.
By Definition 3.2 the statement of the lemma follows easily.
Theorem 3.11.
Let be secondorder generalized differentiable such that is (1,1)differentiable and is (2,1)differentiable or is (1,2)differentiable and is (2,2)differentiable or is (2,1)differentiable and is (1,1)differentiable or is (2,2)differentiable and is (1,2)differentiable on . If the difference exists for then is secondorder generalized differentiable and
for all .
Proof.
We prove the first case and other cases are similar. Since is (1)differentiable and is (2)differentiable on , by [10, Theorem 4], is (1)differentiable and we have . By differentiation as (1)differentiability in Definition 3.2 and using Lemma 3.10, we get is (1,1)differentiable and we deduce
The difference of two functions is understood pointwise.
Theorem 3.12.
Let and be two differentiable functions ( is generalized differentiable as in Definition 3.2).
(i)If and is (1)differentiable, then is (1)differentiable and

(ii)
If and is (2)differentiable, then is (2)differentiable and
Proof.
See [10].
Theorem 3.13.
Let and be secondorder differentiable functions ( is generalized differentiable as in Definition 3.7).
(i)If and is (1,1)differentiable then is (1,1)differentiable and

(ii)
If and is (2,2)differentiable then is (2,2)differentiable and
Proof.
We prove (i), and the proof of another case is similar. If and is (1)differentiable, then by Theorem 3.12 we have
Now by differentiation as first case in Definition 3.2, since is (1)differentiable and then we conclude the result.
Remark 3.14.
By [9, Remark 16], let and define by , for all . If is differentiable on then is differentiable on , with . By Theorem 3.12, if then is (1)differentiable on . Also if then is (2)differentiable on . If , by [9, Theorem 10], we have . We can extend this result to secondorder differentiability as follows.
Theorem 3.15.
Let be twice differentiable on , and define by , for all .
(i)If and then is (1,1)differentiable and its second derivative, is ,

(ii)
If and then is (1,2)differentiable with ,

(iii)
If and then is (2,1)differentiable with ,

(iv)
If and then is (2,2)differentiable with .
Proof.
Cases (i) and (iv) follow from Theorem 3.13. To prove (ii), since , by Remark 3.14, is (1)differentiable and we have on . Also, since , then is (2)differentiable and we conclude the result. Case (iii) is similar to previous one.
Example 3.16.
If is a fuzzy number and where
is crisp secondorder polynomial, then for
we have the following
(i)for: and then by (iv), is (22)differentiable and its second derivative, is ,
(ii)for: and then by (ii), is (12)differentiable with ,
(iii)for: and then by (iii), is (21)differentiable and ,
(iv)for: and then by (i), is (11)differentiable and ,
(v)for: we have , then by [9, Theorem 10] we have , again by applying this theorem, we get