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.