By using the conclusion of uniqueness of meromorphic function and its first and k-th derivative, and combining the conclusion of the uniqueness of algebroid function and its first derivative, the uniqueness theorem of meromorphic function and its k-th derivative studied by Frank and Weissenborn is extended to algebroid function. The results show that if a v-valued algebroid function (v≥2) and its k-th derivative function at least CM share 2v small functions and IM share ∞, then the two are equal. Herefrom, it is inferenced that if a v-valued algebroid function (v≥2) and its k-th derivative function CM share 2v small functions and IM share ∞, then the two are equal. The number of shared values can be reduced to 2v-1, when the v-value is equal to or greater than 3 (v≥3), that is, if an algebroid body function and its derivative functions CM share 2v - 1 finite complex numbers including 0 and IM share ∞, then the two are equal.