Question

How do you implicitly differentiate `-y= x^3y^2-3x^2y^2-7x^2y^4`?

Answer 1

`dy/dx=(x^2y^2-6xy^2-14xy^4)/(28x^2y^3+6x^2y-2x^3y-1)`

Explanation:

This implicit differentiation makes heavy use of the product rule. Before differentiating the function, we should do a sample first:

`d/dx[5x^4y^2]=5y^2d/dx[x^4]+5x^4d/dx[y^2]=5y^2(4x^3)+5x^4(2y)dy/dx`

The most important part about this is that differentiating any term with a `y` will spit out a `dy/dx` term, thanks to the chain rule.

Differentiating the given function:

`-dy/dx=3x^2y^2+2x^3ydy/dx-6xy^2-6x^2ydy/dx-14xy^4-28x^2y^3dy/dx`

Solve for `dy/dx`.

`28x^2y^3dy/dx+6x^2ydy/dx-2x^3ydy/dx-dy/dx=3x^2y^2-6xy^2-14xy^4`

Factor out a `dy/dx` and divide:

`dy/dx=(x^2y^2-6xy^2-14xy^4)/(28x^2y^3+6x^2y-2x^3y-1)`