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Find \(\dfrac{dy}{dx}\) when \(y= x^y + y^x\).

 Aug 31, 2016
 #1
avatar+14995 
0

Hello Max!

 

Find \(\frac{dy}{dx} \) when \(y= x^{y} +y^{x}\)  

 

\(lny=y\times lnx+ x\times lny \)

 

\(lny-x\times lny=y\times lnx\)

 

\(lny\times \left(1-x\right)= y\times lnx \)

 

\(\frac{y}{lny} = \frac{\left(1-x\right) }{lnx} \)

 

\(y=\frac{lny}{lnx} \times \left(1-x\right) \)

 

From here I have to capitulate.
To differentiate, I need
y isolated on one side of the equation.
I'm excited about the solution!

 

Greetings asinus :- ) laugh !

 Aug 31, 2016
edited by asinus  Aug 31, 2016
 #2
avatar+33661 
+1

Not quite asinus!   ln(y) is not equal to y*ln(x) + x*ln(y)

 

Do dy/dx = d(x^y)/dx  +  d(y^x)/dx  first, then treat the terms on the right seperately.

.

 Aug 31, 2016
 #3
avatar+14995 
0

Ojemine which was a m****r errors!

Sorry!

asinus  Aug 31, 2016
 #4
avatar+33661 
+5

As follows:

 

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.

 Aug 31, 2016

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