chapter  16
Higher Order Differentiable Functions
ByRavi P. Agarwal, Peter Cristina, O’Regan Donal
Pages 10

The derivative of a function f is again a function f′. The function f′ may be differentiable, and its derivative we denote by f″ and call it the second derivative of f. In this manner, the function f may be differentiable n ≥ 1 times, and we denote its nth derivative by f. In order for f (n)(x 0) to exist, f (n−1)(x) must exist in an nbd of x 0 (or in a one-sided nbd, if x 0 is an endpoint of the interval on which f is defined), and f ( n−1 must be differentiable at x 0. Since f (n−1) must exist in an nbd of x 0, f (n−2) must be differentiable, and hence continuous, in that nbd. For n ≥ 1, C (n)(I) represents the class of all n times continuously differentiable functions on the interval I. In this chapter, we will prove Taylor′s theorem (generalized mean-value theorem) and illustrate some of its applications.