How do you implicitly differentiate #-3=5x^3y-x^2y+y^2/x#?

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Answer 1 · 2015-11-29T06:52:40

y'=# (y^2+2x^3y-15x^4y)/(5x^5-x^4+2xy)#

#5x^3y-x^2y+y^2/x#=-3
Differentiating on both sides with respect to x

d/dx#( 5x^3y)-d/dx(-x^2y)+d/dx(y^2/x)#=d/dx(-3)

Use product rule for first two and quotient rule for third part

#15x^2y+5x^3y'-2xy-x^2y'+(2yy'x-y^2)/x^2#=0

#(15x^4y+5x^5y'-2x^3y-x^4y'+2yy'x-y^2)/x^2#=0

A rational expression is 0 , only if the numerator is 0

so #(15x^4y+5x^5y'-2x^3y-x^4y'+2yy'x-y^2)#=0

solve for y'

#(5x^5-x^4+2xy)y'= y^2+2x^3y-15x^4y#

y'=# (y^2+2x^3y-15x^4y)/(5x^5-x^4+2xy)#