Problem:Use implicit differentiation to compute ∂ z ∂ x and ∂ z ∂ y of the function x 3 + y 3 + z 3 − 3 x y z = 0.What I Got: 3 x 2 + 0 + 3 z 2 ⋅ ∂ z ∂ x − 3 y ⋅ ∂ z ∂ x ∂ z ∂ x ( 3 z 2 − 3 y ) = − 3 x 2 ∂ z ∂ x = − 3 x 2 3 z 2 − 3 y and 0 + 3 y 2 + 3 z 2 ⋅ ∂ z ∂ y − 3 x ⋅ ∂ z ∂ y ∂ z ∂ y ( 3 z 2 − 3 x ) = − 3 y 2 ∂ z ∂ y = − 3 y 2 3 z 2 − 3 y For what I have got, is this the correct way to do this question?

Question

Problem:Use implicit differentiation to compute             ∂      z              ∂      x       and             ∂      z              ∂      y       of the function       x    3    +      y    3    +      z    3    −  3  x  y  z  =  0.What I Got:  3      x    2    +  0  +  3      z    2    ⋅            ∂      z              ∂      x        −  3  y  ⋅            ∂      z              ∂      x                  ∂      z              ∂      x        (  3      z    2    −  3  y  )  =  −  3      x    2              ∂      z              ∂      x        =            −      3              x        2                    3              z        2            −      3      y      and  0  +  3      y    2    +  3      z    2    ⋅            ∂      z              ∂      y        −  3  x  ⋅            ∂      z              ∂      y                  ∂      z              ∂      y        (  3      z    2    −  3  x  )  =  −  3      y    2              ∂      z              ∂      y        =            −      3              y        2                    3              z        2            −      3      y      For what I have got, is this the correct way to do this question?

Guadalupe Reid · Accepted Answer

You have made an error in implicit partial differentiation of the   −  3  x  y  z term. Note that            ∂      (      −      3      x      y      z      )              ∂      x        =  −  3  y            ∂      (      x      z      )              ∂      x      since   y is treated as a constant when differentiating with respect to   x. Recall that   z is actually a function of both   y and   x, i.e.   z  =  z  (  x  ,  y  ), so   z can&#039;t be given the same treatment as   y.You can evaluate this using the product rule of differentiation, i.e.   (  u  v      )    x    =      u    x    v  +  u      v    x  , to get  −  3  y            ∂      (      x      z      )              ∂      x        =  −  3  y      [    z    +    x                  ∂        z                    ∂        x              ]  Similarly when you are differentiating with respect to   y.

Use implicit differentiation to compute (del z)/(del x) and (del z)/(del y) of the function x^3+y^3+z^3−3xyz=0.

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