Question

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

Answer 1

`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