d z d x = ? c o s ( z y ) + z x 2 = ( 1 + y ) e ( x − z )For the left side through implicit differentiation I have found ( − s i n ( z y ) ) ( y ∗ ( d z / d x ) ) + 2 x z + ( d z / d x ) x 2 . I am completely unsure how to approach the right side, however.

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d      z              d      x        =  ?  c  o  s  (  z  y  )  +  z      x    2    =  (  1  +  y  )      e    (    x  −  z  )For the left side through implicit differentiation I have found   (  −  s  i  n  (  z  y  )  )  (  y  ∗  (  d  z      /    d  x  )  )  +  2  x  z  +  (  d  z      /    d  x  )      x    2  . I am completely unsure how to approach the right side, however.

Karli Moreno · Accepted Answer

The problem is quite simple if you use the implicit function theorem.Consider the implicit function  F  =  cos  ⁡  (  y  z  )  +      x    2    z  −  (  y  +  1  )      e          x      −      z            =    0  Then            ∂      F              ∂      x        =  2  x  z  −  (  y  +  1  )      e          x      −      z                  ∂      F              ∂      z        =      x    2    +  (  y  +  1  )      e          x      −      z        −  y  sin  ⁡  (  y  z  )            ∂      z              ∂      x        =  −                    ∂        F                    ∂        x                            ∂        F                    ∂        z              =            (      y      +      1      )              e                  x          −          z                    −      2      x      z                      x        2            +      (      y      +      1      )              e                  x          −          z                    −      y      sin      ⁡      (      y      z      )

dz/dx = ? cos(zy)+zx^2 = (1+y)e^(x-z) For the left side through implicit differentiation I have found (-sin(zy))(y*(dz/dx))+2xz+(dz/dx)x^2. I am completely unsure how to approach the right side, however.

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