Question

How do you determine dy/dx given `y=x^2+xy`?

Answer 1

Given a choice, I would solve for `y` first.

Explanation:

`y=x^2+xy`

Making the function explicit
`y-xy=x^2`

`y = x^2/(1-x)`

`dy/dx = (2x(1-x^2)-x^2(-1))/(1-x)^2`

`= (2x-x^2)/(1-x)^2`

Leaving the function implicit

`y=x^2+xy`

`d/dx(y)=d/dx(x^2+xy)`

`dy/dx = 2x +y + x dy/dx`

`dy/dx = (2x+y)/(1-x)`

Answer 2

`dy/dx=(2x+y)/(1-x)`

Explanation:

You have to use Implicit Differentiation, which is a special case of the chain rule, and the product rule.

`dy/dx=2x+x(1)dy/dx+y(1)`

Simplify

`dy/dx=2x+xdy/dx+y`

Solve for `dy/dx`

`dy/dx-xdy/dx=2x+y`

Factor out `dy/dx`

`dy/dx(1-x)=2x+y`

Isolate `dy/dx`

`(dy/dxcancel(1-x))/(cancel(1-x))=(2x+y)/(1-x)`

Simplify

`dy/dx=(2x+y)/(1-x)`

Here is an example of using the implicit differentiation.