I am trying to solve the problem: x 2 + x y + y 3 = 0 using implicit differentiation.My workings: ( 1 ) d d x [ x 2 ] + d d x [ x y ] + d d x [ y 3 ] = d d x [ 0 ] ( 2 ) 2 x + y + d y 3 d y d y d x = 0 ( 3 ) 2 x + y + 3 y 2 ( d y d x ) = 0 ( 4 ) d y d x = − 2 x + y 3 y 2 But the answer says it should be: ( 3 ) 2 x + y + d x y d y d y d x + 3 y 2 ( d y d x ) = 0 ( 4 ) 2 x + y + d y d x ( x + 3 y 2 ) = 0 ( 5 ) d y d x = − 2 x + y x + 3 y 2 Why?

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I am trying to solve the problem:       x          2        +  x  y  +      y          3        =  0 using implicit differentiation.My workings:  (  1  )                  d              d        x                  [          x      2        ]      +            d              d        x                  [    x    y    ]      +              d              d        x                  [          y      3        ]    =            d              d        x                  [    0    ]    (  2  )        2  x  +  y  +                    d                  y          3                            d        y                                d        y                    d        x              =  0  (  3  )        2  x  +  y  +  3      y    2      (                    d        y                    d        x              )  =  0  (  4  )                          d        y                    d        x              =                              −                                                    2                x                +                y                                            3                                  y                  2                                                                        But the answer says it should be:  (  3  )        2  x  +  y  +                    d        x        y                    d        y                                d        y                    d        x              +  3      y    2      (                    d        y                    d        x              )  =  0  (  4  )        2  x  +  y  +                    d        y                    d        x              (  x  +  3      y    2    )  =  0  (  5  )                          d        y                    d        x              =  −                    2        x                +                y                    x                +                3                  y          2                    Why?

Barbara Klein · Accepted Answer

You can&#039;t conclude that       d          d      x        (  x  y  )  =  y, since this is effectively the product of the functions   x and   y. Hence, using the product rule gives      d          d      x        (  x  y  )  =  y      d          d      x        (  x  )  +  x      d          d      x        (  y  )  =  y  +  x            d      y              d      x      The rest should work out as expected.

Taliyah Reyes · Accepted Answer

Differentials will make the process much more intuitive.                          0        =        d                              (                                    x          2                +        x        y        +                  y          3                                      )                                                    0        =        2        x                d        x        +                              (                          x                d        y        +        y                d        x                              )                          +        3                  y          2                d        y                                                        (                          x        +        3                  y          2                                      )                                  d        y        =        −                              (                          2        x        +        y                              )                                  d        x                                                        d            y                                d            x                          =        −                  (                                    2              x              +              y                                      x              +              3                              y                2                                              )

I am trying to solve the problem: x^2+xy+y^3=0 using implicit differentiation.

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