How do you implicitly differentiate #-y= 4x^3y^2+2x^2y^3-2xy^4 #?

1 Answer
Oct 6, 2016

#y'=(-12x^2y^3-4xy^3+2y^4)/(1+8x^3y+6x^2y^2-8xy^3)#

Explanation:

Implicit Differentiation is a special case of the chain rule. When you differentiate the y variable in a term or factor you differentiating with respect to x. Because of this you have include the factor of #dy/dx# or #y'#.

We will have to use the product rule, power rule and chain rule.

Divide all the terms by -1 to remove the negative

#y=-4x^3y^2-2x^2y^3+2xy^4#

Differentiate Implicitly

#y'=-4x^3 2yy'-12x^2y^3-(2x^2 3y^2 y'+4xy^3)+2x4y^3y'+2y^4#

Distribute the negative

#y'=-4x^3 2yy'-12x^2y^3-2x^2 3y^2 y'-4xy^3+2x4y^3y'+2y^4#

Gather the terms with the #y'# factors

#y'+4x^3 2yy'+2x^2 3y^2 y'-2x4y^3y'=-12x^2y^3-4xy^3+2y^4#

Factor out #y'#

#y'(1+4x^3 2y+2x^2 3y^2-2x4y^3)=-12x^2y^3-4xy^3+2y^4#

Divide to isolate #y'#

#(y'cancel((1+4x^3 2y+2x^2 3y^2-2x4y^3)))/cancel((1+4x^3 2y+2x^2 3y^2-2x4y^3))=(-12x^2y^3-4xy^3+2y^4)/(1+4x^3 2y+2x^2 3y^2-2x4y^3)#

#y'=(-12x^2y^3-4xy^3+2y^4)/(1+4x^3 2y+2x^2 3y^2-2x4y^3)#

Multiple numeric factors to simplify

#y'=(-12x^2y^3-4xy^3+2y^4)/(1+8x^3y+6x^2y^2-8xy^3)#

Below are a couple of tutorials include implicit differentiation