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!

Deanna Gregory 2022-09-28 Answered
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!
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Answers (2)

Lichtpulsax
Answered 2022-09-29 Author has 4 answers
Consider the implicit function
f ( K , L ) = e L + K L K e K = 0
Now
f ( K , L ) K = L e K ( K + 1 ) and f ( K , L ) L = e L + K
Now, 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
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emmostatwf
Answered 2022-09-30 Author has 2 answers
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 K
now 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
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