Find dy/dx by implicit differentiation : e^(x/y) = x - y

Question

Find dy/dx by implicit differentiation : e^(x/y) = x - y

Answer 1

Ok, so we have have to take the derivative of both sides.

 

on the left we need to use the quotient rule 

 

u=e^x             v = y

u'=e^x            v'= dy/dx

 

putting that together we get

 

y*e^x-e^x dy/dx/y^2  

 

on the right we get 

 

-dy/dx

 

so all together

 

y*e^x-e^x dy/dx/y^2 = -dy/dx

 

we need to get all the dy/dx terms onto one side so we will have 

 

y*e^x= e^x dy/dx - y^2 dy/dx

 

factoring 

 

y*e^x = dy/dx (e^x - y^2)

 

Lastly 

 

dydx = y*e^x/(e^x - y^2)