xlog(x)+ylog(y)=1 dy/dx=? Implicit Differentiation - Logarithm

Barrett Osborn

Barrett Osborn

Answered question

2022-11-06

x log ( x ) + y log ( y ) = 1
d y d x = ?
I calculated d y d x = 1 + log ( x ) 1 + log ( y )
however, the correct answer seems to be log ( x ) / log ( y )
I'm confused, can someone help?

Answer & Explanation

erlentzed

erlentzed

Beginner2022-11-07Added 22 answers

Assuming that log ( x ) = log e ( x ) = ln ( x ), we start off with our equation:
x log ( x ) + y log ( y ) = 1
Next we perform implicit differentiation to both sides:
d y d x ( x log ( x ) + y log ( y ) ) = d y d x ( 1 ) 1 + log ( x ) + d y d x 1 + d y d x log ( y ) = 0
Now we isolate d y d x to one side:
1 + log ( x ) + d y d x 1 + d y d x log ( y ) = 0 d y d x 1 + d y d x log ( y ) = log ( x ) 1 d y d x ( 1 + log ( y ) ) = log ( x ) 1 d y d x = log ( x ) 1 1 + log ( y )
Finally we clean things up and simplify:
d y d x = log ( x ) 1 1 + log ( y ) = 1 + log ( x ) 1 + log ( y )
We see that your solution is indeed correct.
spasiocuo43

spasiocuo43

Beginner2022-11-08Added 6 answers

I got
log ( x ) + 1 + y log ( y ) + y = 0

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