How to find ((d^2)y)/(dx^2)?

Question

Camuccinirk84 · Accepted Answer

Defining                 f                (         x         ,         y                    (           x           )                  )                 =     2     x     y            (       x       )          +     2     y                     (         x         )              2          -     13     =     0     then                         d         f                                  d           x                          =     2     y     +     2     x     y     &amp;#x2032;     +     4     y     y     &amp;#x2032;     =     0     and                d                           d           x                                          d         f                                  d           x                          =     4     y     &amp;#x2032;     +     2     x     y     &amp;#x2032;     &amp;#x2032;     +     4     y            &amp;#x2032;       2          +     4     y     y     &amp;#x2032;     &amp;#x2032;     =     0     after substitution of         y     &amp;#x2032;     we have         y     &amp;#x2032;     &amp;#x2032;     =                     2         x         y         +         2                    y           2                                           (           x           +           2           y           )                  3                 =            13                           (           x           +           2           y           )                  3                 but         y     =            1       2                 (       -       x       &amp;#xB1;                           26           +                        x             2                                  )          so finally         y     &amp;#x2032;     &amp;#x2032;     =     &amp;#xB1;            13                           (           26           +                        x             2                      )                                          3             2

Ruby Rollins · Accepted Answer

2     x     y     +     2            y       2          =     13     Differentiating wrt         x     and applying the product rule gives us:         2            {                (         x         )                       (                                 d             y                                   d             x                             )              +                (         1         )                       (         y         )              }          +     4     y                     d         y                       d         x                 =     0             x                     d         y                       d         x                 +     y     +     2     y                     d         y                       d         x                 =     0     &amp;#x21D2;                     d         y                       d         x                 =     -            y                x         +         2         y                 Differentiating again wrt         x     and applying the product rule (twice) gives us:         &amp;#x2234;            {                (         x         )                       (                                                d               2                          y                                                                   d                 x                              2                                          )              +                (         1         )                       (                                 d             y                                   d             x                             )              }          +                     d         y                       d         x                 +     2            {                (         y         )                       (                                                d               2                          y                                                                   d                 x                              2                                          )              +                (         2                                 d             y                                   d             x                             )                       (                                 d             y                                   d             x                             )              }          =     0             &amp;#x2234;     x                                d           2                  y                                               d             x                      2                          +                     d         y                       d         x                 +                     d         y                       d         x                 +     2     y                                d           2                  y                                               d             x                      2                          +     2                     (                                 d             y                                   d             x                             )              2          =     0             &amp;#x2234;     x                                d           2                  y                                               d             x                      2                          +     2                     d         y                       d         x                 +     2     y                                d           2                  y                                               d             x                      2                          +     2                     (                                 d             y                                   d             x                             )              2          =     0             &amp;#x2234;            (       x       +       2       y       )                                     d           2                  y                                               d             x                      2                          +     2            (       -                y                    x           +           2           y                       )          +     2                     (         -                    y                        x             +             2             y                             )              2          =     0             &amp;#x2234;            (       x       +       2       y       )                                     d           2                  y                                               d             x                      2                          =                     2         y                       x         +         2         y                 -                     2                    y           2                                           (           x           +           2           y           )                  2                         &amp;#x2234;                                d           2                  y                                               d             x                      2                          =                     2         y                                  (           x           +           2           y           )                  2                 -                     2                    y           2                                           (           x           +           2           y           )                  3                  Overlaying a common denominator results in:                                     d           2                  y                                               d             x                      2                          =                                (           2           y           )                             (           x           +           2           y           )                  -                    (           2                        y             2                      )                                           (           x           +           2           y           )                  3                          &amp;#x2234;                                d           2                  y                                               d             x                      2                          =                                (           2           x           y           +           4                        y             2                      -           2                        y             2                      )                                           (           x           +           2           y           )                  3                          &amp;#x2234;                                d           2                  y                                               d             x                      2                          =                                (           2           x           y           +           2                        y             2                      )                                           (           x           +           2           y           )                  3                          &amp;#x2234;                                d           2                  y                                               d             x                      2                          =            13                           (           x           +           2           y           )                  3                 (as         2     x     y     +     2            y       2          =     13    )

How to find ((d^2)y)/(dx^2)?

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