How do you use implicit differentiation to find #(dy)/(dx)# given #x^2+xy+y^2=9#?

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Answer 1 · 2017-01-02T12:39:20

#y' = -2(x+y)/(x+2y)#

Differentiate the equation with respect to #x#, considering that based on the chain rule:

#d/(dx) f(y) = (df)/(dy)*(dy)/(dx)#

Thus we have:

#d/(dx) (x^2+xy+y^2) = 0#

#2x+ xy'+y+2yy'=0#

solving for #y'#:

#y'(x+2y) = -2x-y#

#y' = -2(x+y)/(x+2y)#