How do you implicitly differentiate #x^2 y^3 − xy = 10#?

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Answer 1 · 2016-03-08T18:18:56

You can do it like this

#x^2y^3-xy=10#

Differentiate both sides with respect to #x#:

#D(x^2y^3-xy)=D(10)#

Then apply the product rule and the chain rule:

#x^(2)3y^2y'+y^(3)2x-(xy'+y)=0#

#3x^(2)y^(2)y'+2y^(3)x-xy'-y=0#

#:.y'(3x^2y^2-x)=y-2y^3x#

#:.y'(3x^2y^2-x)=y(1-2xy^2)#

#:.y'=(y(1-2xy^2))/((3x^2y^2-x))#

#y'=(y(1-2xy^2))/(x(3xy^2-1)#