SOLUTION: If ln(lny) + lny = lnx, find y'

Question 1119541: If ln(lny) + lny = lnx, find y'

Found 2 solutions by Tatiana_Stebko, greenestamps:
Answer by Tatiana_Stebko(1539) (Show Source): You can put this solution on YOUR website!
ln(lny) + lny = lnx
Differentiate both sides of the equation
(ln(lny) + lny)' =( lnx)'
(lny)'/lny + y'/y =1/x
y'/(y*lny) + y'/y =1/x
Solve for y'
y'(1/(y*lny) + 1/y)=1/x
y'((1+lny)/(y*lny)) =1/x
y' =(y*lny)/(x(1+lny))

Answer by greenestamps(13203) (Show Source): You can put this solution on YOUR website!

The solution by the other tutor can be simplified; but it is difficult to see how.

The simpler form of the answer can be obtained more easily via a different path.

ln(ln(y))+ln(y) = ln(x)

Write the expression on the left as a single logarithm:

ln(y*ln(y)) = ln(x)

y*ln(y) = x

Now do the differentiation using the product rule:

y'(ln(y))+y(1/y)y' = 1
y'(ln(y))+y' = 1
y'(1+ln(y)) = 1
y' = 1/(1+ln(y))

The answer from the other tutor was this:

y' =(y*ln(y))/(x(1+ln(y)))

This is equivalent to

y' = 1/(1+ln(y))

because

y*ln(y) = x
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