Given e L + K L = K e K , we are being asked to find d L d K . I think O need to use implicit differentiation, but I am really not sure how to do it!

Question

Given       e    L    +  K  L  =  K      e    K  , we are being asked to find             d      L              d      K      . I think O need to use implicit differentiation, but I am really not sure how to do it!

Lichtpulsax · Accepted Answer

Consider the implicit function  f  (  K  ,  L  )  =      e    L    +  K  L  −  K      e    K    =  0Now            ∂      f      (      K      ,      L      )              ∂      K        =  L  −      e    K    (  K  +  1  )    and              ∂      f      (      K      ,      L      )              ∂      L        =      e    L    +  KNow, using the implicit function theorem            d      L              d      K        =  −                    ∂        f        (        K        ,        L        )                    ∂        K                            ∂        f        (        K        ,        L        )                    ∂        L              =                    e        K            (      K      +      1      )      −      L              K      +              e        L

emmostatwf · Accepted Answer

There are a lot of different ways to do implicit differentiation. Some methods have you differentiate respect to a particular variable, but, the way I prefer is to just do differentiation without being with respect to a particular variable, and then solve for the derivative you are looking for. So, instead of the derivative rule for   y  =      u    n   being             d      y              d      u        =  n      u          n      −      1      , I instead have the differential rule   d  y  =  n      u          n      −      1          d  u. Then I can algebraically divide by   d  u if I want to find             d      y              d      u      .So, in your case, we have:      e    L    +  K  L  =  K      e    K  Differentiating both sides gives us:  d      (          e      L        +    K    L    )    =  d      (    K          e      K        )      d      (          e      L        )    +  d      (    K    L    )    =  d      (    K          e      K        )          e    L      d  L  +  K    d  L  +  L    d  K  =  K      e    K      d  K  +      e    K      d  Know we just need to gather our terms together so we can solve for             d      L              d      K      :      e    L      d  L  +  K    d  L  +  L    d  K  =  K      e    K      d  K  +      e    K      d  K        e    L      d  L  +  K    d  L  =  K      e    K      d  K  +      e    K      d  K  −  L    d  K        (          e      L        +    K    )      d  L  =      (    K          e      K        +          e      K        −    L    )      d  K              d      L              d      K        =            K              e        K            +              e        K            −      L                      e        L            +      K

Given e^L+KL=Ke^K, we are being asked to find (dL)/(dK). I think O need to use implicit differentiation, but I am really not sure how to do it!

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