The quotient of a formal language K by another language L is the set of all strings obtained by taking a string from K that ends with a suffix of a string from L, and removing that suffix. The quotient of a regular language by any language is always regular, whereas the context-free languages and many of their subfamilies, such as the linear and the deterministic languages, are not closed under the quotient operation. This paper establishes the closure of the family of languages recognized by input-driven pushdown automata (IDPDA), also known as visibly pushdown automata, under the quotient operation. A construction of automata representing the result of the operation is given, and its state complexity with respect to nondeterministic IDPDA is shown to be exactly m2n, where m and n are the numbers of states in the automata recognizing K and L, respectively.