How to find y'' by implicit differentiation for 4 x 2 + 3 y 2 = 6 ?

Question

How to find y&#039;&#039; by implicit differentiation for       4          x      2        +    3          y      2        =    6  ?

Giovanna Shaw · Accepted Answer

4          x      2        +    3          y      2        =    6              d                        d          x                            (      4              x        2            +      3              y        2            )        =          d                        d          x                            (      6      )            8    x    +    6    y                            d          y                                      d          x                      =    0  So                                d          y                                      d          x                      =                  -        4        x                    3        y            The second derivative follows:            d                        d          x                            (                                    d            y                                                d            x                              )        =          d                        d          x                            (                        -          4          x                          3          y                    )                                    d          2                y                                          d            x                    2                      =                            (          -          4          )                          (          3          y          )                -                  (          -          4          x          )                          (          3                                                    d                y                                                    d              x                                )                                                  (            3            y            )                    2                                                  d          2                y                                          d            x                    2                      =                  -        12        y        +        12        x                                            d              y                                            d            x                                      9                  y          2                      Replacing                     d        y                    d        x             by the expression above gives us:                              d          2                y                                          d            x                    2                      =                  -        12        y        +        12        x                  (                                    -              4              x                                      3              y                                )                            9                  y          2                      =                  -        12        y        +                              -            16                          x              2                                y                            9                  y          2                      No multiply by             y      y       to get                               d          2                y                                          d            x                    2                      =                  -        12                  y          2                -        16                  x          2                            9                  y          3                      =                  -        4                  (          4                      x            2                    +          3                      y            2                    )                            9                  y          3                    Now, use the initial equation:        4          x      2        +    3          y      2        =    6    to write the answer as:                              d          2                y                                          d            x                    2                      =    =                  -        24                    9                  y          3                      =                  -        8                    3                  y          3

How to find y'' by implicit differentiation for 4 x 2 + 3 y 2...

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