How do you implicitly differentiate #-3=(xy)/(x-e^y)#?

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Answer 1 · 2016-12-23T21:48:37

#(dy)/(dx)=1/x#

Let's write the equation in the form:

#-3x+3e^y=xy#

and differentiate term to term with respect to #x#:

#-3+3e^y(dy)/(dx) = y#

no solve for the derivative:

#(dy)/(dx)= (y+3)/(3e^y)#

From the initial equation we can derive:

#3e^y=xy+3x=x(y+3)#

and substituting this in the expression of the derivative we get:

#(dy)/(dx)= (y+3)/(3e^y)=(y+3)/(x(y+3))=1/x#