There is no trouble when I have to differentiate a function f ( x , y ) where y = g ( x ) using Implicit Function Theorem. It's easy to get g x x and g y y . For instance, the formula for g x x = − f x x F y 2 − ( F x y + F y x ) F x F y + F y y F x 2 F y 3 knowing that g ′ ( x ) = − F x F y whenever F y ≠ 0 .Now, i'm in the situation z + x + ( y + z ) 4 = 0 where it defines an implicit function z = f ( x , y ).I got f x and f y using the theorem. However, I can't get f x x and f y y from there. I know I shoud apply the chain rule but I get lost whenever trying to get a "general formula" for them.

Question

There is no trouble when I have to differentiate a function   f  (  x  ,  y  ) where   y  =  g  (  x  ) using Implicit Function Theorem. It&#039;s easy to get       g          x      x       and       g          y      y      . For instance, the formula for       g          x      x        =  −                    f                  x          x                            F        y        2            −      (              F                  x          y                    +              F                  y          x                    )              F        x                    F        y            +              F                  y          y                            F        x        2                    F      y      3       knowing that       g    ′    (  x  )  =  −            F      x              F      y       whenever       F    y    ≠      0  .Now, i&#039;m in the situation   z  +  x  +  (  y  +  z      )    4    =  0 where it defines an implicit function   z  =  f  (  x  ,  y  ).I got       f    x   and       f    y   using the theorem. However, I can&#039;t get       f          x      x       and       f          y      y       from there. I know I shoud apply the chain rule but I get lost whenever trying to get a &quot;general formula&quot; for them.

sviudes7w · Accepted Answer

From   F  (  x  ,  y  ,  z  (  x  ,  y  )  )  =  0,                                    ∂                      ∂            x                          F        =                  F          x                +                  F          z                          z          x                =        0                ⟹                          z          x                =        −                              F            x                                F            z                                                                          ∂            2                                ∂                          x              2                                      F        =                  F                      x            x                          +                  F                      x            z                                    z          x                +                  z          x                (                  F                      z            x                          +                  F                      z            z                                    z          x                )        +                  F          z                          z                      x            x                          =        0            Plugging in       z    x   and rearranging, it should be clear how to solve for       z          x      x      . A similar method holds for   y.