An algorithm of the calculation of derivatives of an implicit function

A method of the formalization of the expression for high derivatives of an implicit function is suggested. An algorithm of the calculation of these expressions by the computer is constructed. As an example, the equation $J_{\nu}(x)=0$ is considered where $J_{\nu}(x)$ is the Bessel function of index $\nu$; its solutions $\nu=\nu(x)$ are approximated by the Taylor's polynomial. The coefficients of the approximation are calculated for the first five zeros and the precision of the approximating formulas is examined numerically.