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¡¡¡¡the implicit function theorem is. along with its close cousin the inverse func- tion theorem£¬ one of the most important£¬ and one of the oldest£¬ paradigms in modcrn mathemarics. one can see the germ of the idea for the implicir func tion theorem in the writings of isaac newton £¨1642-1727£©£¬ and gottfried leib-niz's £¨1646-1716£© work cxplicitty contains an instance of implicit differentiation. ¡¡¡¡whilc joseph louis lagrange £¨1736-1813£© found a theorcm that is essentially a version of the inverse function theorem£¬ ic was augustin-louis cauchy £¨1789-1857£© who approached the implicit function theorem with mathematical rigor and it is he who is gencrally acknowledgcd as the discovcrer of the theorem. in chap-ter 2£¬ we will give details of the contributions of newton£¬ lagrange£¬ and cauchy to the development of the implicit function theorem.