Two mathematical derivations of them have been introduced by using the properties of differentiable function and the derivation algorithm of multiple function in present paper, and several mistakes in some derivative methods have also been pointed out.

In this paper, we give a generic condition of differentiable functions on noncompact complete Rie- mannian manifold.

This paper investigates the absolute convergence of the Fourier Laplace series concerning of some smooth functions defined on the unit sphere in R n,hereinto shows that:if f is 2([n4]+1) th continuously differentiable function on H r P(Ω n),then the series ∑∞k=0Y kf(x) converges uniformly to f.

In this paper,by using the result about the simultaneous polynomial approximation with Hermite interpolatory side conditions,we discuss general conformal simultaneous approximation of differentiable functions and obtain some results.

In the paper,some regularization properties of ω-ultradifferentiable functions D(w)(Rn) and ω-ultradistributions D′(w)(Rn) are discussed.

In the paper,a special class of quasidifferentiable functions-subsuper differentiable functions is introduced,which satisfies the conditions of the existence of high dimensional kernels,and the formula of kernels is deduced.

In this paper, the convexificator kernels of quasidifferentiable functions are introduced.

An algorithm for determining the Gteax differentiability for a class of quasidifferentiable functions whose quasidifferentials are convex hulls of finite number of points is presented.

Considiering the following indifferentiable optimization problemmin where arequasidifferentiable functions (in the sense of Demyanov and Rubinov)defined onRn and are locally Lipschitz defined on Rn.

A characterization on quasidifferentiable functions is given this paper.