Let z = z ( x , y ) be defined implicitly by F ( x , y , z ( x , y ) ) = 0, where F is a given function of three variables.Prove that if z ( x , y ) and F are differentiable, then d z d x = − d F d x d F d z if d F / d z ≠ 0I am kind of confused with z = z ( x , y ). Should I differentiate d z/ d x and d z/ d y, but there's nothing inside x and y so what should I differentiate d x and d y with then? I've been told that this is something related to implicit differentiation. In some other examples, they do use differentiate F i.e. d F but in most examples, I only see that they are differentiating the variables inside it.

Question

Let   z  =  z  (  x  ,  y  ) be defined implicitly by   F  (  x  ,  y  ,  z  (  x  ,  y  )  )  =  0, where   F is a given function of three variables.Prove that if   z  (  x  ,  y  ) and   F are differentiable, then            d      z              d      x        =  −                                          d            F                                d            x                                                d            F                                d            z                              if   d  F      /    d  z  ≠  0I am kind of confused with   z  =  z  (  x  ,  y  ). Should I differentiate   d  z/  d  x and   d  z/  d  y, but there&#039;s nothing inside   x and   y so what should I differentiate   d  x and   d  y with then? I&#039;ve been told that this is something related to implicit differentiation. In some other examples, they do use differentiate   F i.e.   d  F but in most examples, I only see that they are differentiating the variables inside it.

g5riem7z · Accepted Answer

Consider the function   φ    :    x    ⟼    F  (  x  ,  y  ,  z  (  x  ,  y  )  ). We have :                              (                      ⋆                    )                            φ        (        x        )        =        0            Differentiate (using the chain rule) this equality with respect to   x. You will get:      φ    ′    (  x  )  =            ∂      F              ∂      x                  (        x  ,  y  ,  z  (  x  ,  y  )            )        +            ∂      z              ∂      x        (  x  ,  y  )            ∂      F              ∂      z                  (        x  ,  y  ,  z  (  x  ,  y  )            )        =  0.As a consequence, if                     ∂        F                    ∂        z                            (              x    ,    y    ,    z    (    x    ,    y    )                  )              ≠    0  , we have :            ∂      z              ∂      x        (  x  ,  y  )  =  −                              ∂          F                          ∂          x                                      (                    x      ,      y      ,      z      (      x      ,      y      )                        )                                              ∂          F                          ∂          z                                      (                    x      ,      y      ,      z      (      x      ,      y      )                        )                      .

Let z=z(x,y) be defined implicitly by F(x,y,z(x,y))=0, where F is a given function of three variables. Prove that if z(x,y) and F are differentiable, then (dz)/(dx)=−((dF)/(dx))/((dF)/(dz)) if dF/dz!=0

Open question

Answer & Explanation

New Questions in Calculus 2