I was wondering whether anybody could explain how you derive this implicit differentiation rule: ∂ z ∂ x = − ∂ f / ∂ x ∂ f / ∂ z if you have a function z implicity defined by f ( x , y , z ) = 0?

Question

I was wondering whether anybody could explain how you derive this implicit differentiation rule:            ∂      z              ∂      x        =            −      ∂      f                        /                    ∂      x              ∂      f                        /                    ∂      z      if you have a function   z implicity defined by   f  (  x  ,  y  ,  z  )  =  0?

abortargy · Accepted Answer

Take   z a function of   x  ,  y. Then it is always true that  f  (  x  ,  y  ,  z  (  x  ,  y  )  )  =  0Its derivative is also 0. So            ∂      f              ∂      x        d  x  +            ∂      f              ∂      x        d  y  +            ∂      f              ∂      z        ⋅      (                  ∂        z                    ∂        x              d    x    +                  ∂        f                    ∂        x              d    y    )    =  0.If we consider the case   d  y  =  0,   d  y  ≠  0, then divide by   d  x and cross out   d  y,            ∂      f              ∂      x        +            ∂      f              ∂      z                  ∂      z              ∂      x        =  0.Or,            ∂      z              ∂      x        =  −            ∂      f              /            ∂      x              ∂      f              /            ∂      z

I was wondering whether anybody could explain how you derive this implicit differentiation rule: (del z)/(del x)=−(del f/ del x)/(del f/ del z) if you have a function z implicity defined by f(x,y,z)=0?

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