Page 183 - Foundations Of Differential Calculus
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166    8. On the Higher Differentiation of Differential Formulas
            3
        for d y. In this way the formula
                                      2    2    3/2
                                 − dx + dy
                                                ,
                                         2
                                     dx d y
        in which dx is set constant, is transformed into
                                     2    2    3/2
                                   dx + dy
                                               ,
                                         2
                                     dy d x
        in which dy is set constant.
        280. If, on the other hand, a formula in which dy is set constant is to be
                                                             2
        transformed into another in which dx is constant, then for d x we have to
        substitute
                                          2
                                     −dx d y
                                            ,
                                       dy
                3
        and for d x the expression
                                     2 2
                                              3
                                3dx d y  −  dx d y  .
                                   dy 2     dy
                                              2     2
        In a similar way, if a formula in which  dx + dy is set constant is to be
                                                             2
        transformed into another in which dx is constant, then for d x we write
                                           2
                                   −dx dy d y  ,
                                      2
                                    dx + dy 2
                2
        and for d y we write
                                       2 2
                                     dx d y  .
                                      2
                                    dx + dy 2
        However, if a formula in which dx is assumed to be constant is to be
                                                  2
                                            2

        transformed into another in which  dx + dy is to be constant, since
                                       2
                2
                                               2
          2
        dx + dy is constant, we have dx d x + dy d y = 0 and
                                             2
                                   2     dy d y
                                  d x = −      .
                                           dx
                             2
                                        2
        This value is given to d x, and for d y we write
                                  2 2       2    2    2
                           2    dy d y    dx + dy  d y
                          d y +       =                .
                                 dx 2         dx 2
        Hence the formula
                                      2    2    3/2
                                 − dx + dy      ,
                                         2
                                     dx d y
                                                                  2     2
        in which dx is constant, is transformed into another in which  dx + dy
        is set constant, which is
                                         2

                                 −dx   dx + dy 2  .
                                        2
                                       d y
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