Question

How do you find `(dy)/(dx)` given `-4x^2y^3+2=5x^2+y^2`?

Answer 1

`y'=(8xy^3+10x)/(-4x^2-3y^2-2y)`

Explanation:

Remember that Implicit Differentiation is really just a special case of the Chain Rule.

Every time that we differentiate the a factor or term what includes the variable y we have to include a factor of `dy/dx` or `y'`.

• For the first term, `-4x^2y^3`, we have to use the Product Rule and Power Rule .
• For the constant, `2`, we have to use the Constant Rule .
• For the term, `5x^2`, use the Power Rule .
• For the term, `y^2`, use the Power Rule .

`-4x^2 3y^2y'+(-8)xy^3+0=10x+2yy'`

Gather the terms with `y'` on one side of the equations and other terms on the other side.

`-4x^2 3y^2y'-2yy'=8xy^3+10x`

Factor out `y'`

`y'(-4x^2 3y^2-2y)=8xy^3+10x`

Isolate `y'` by dividing both sides by `(-4x^2 3y^2-2y)`

`y'cancel(-4x^2 3y^2-2y)/cancel(-4x^2 3y^2-2y)=(8xy^3+10x)/(-4x^2 3y^2-2y)`

`y'=(8xy^3+10x)/(-4x^2 3y^2-2y)`

I have a couple of tutorials on Implicit Differentiation here, https://www.youtube.com/playlist?list=PLsX0tNIJwRTxL9RSJY4wKpW1MFbQfA84w