Higher Order Differentiable Functions

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.