Question

How do you implicitly differentiate `y^2+(y-x)^2-y/x^2-3y`?

Answer 1

First expand the brackets to give:
`y^2+(y-x)^2-y/x^2-3y`
`=y^2+y^2-2xy+x^2-y/x^2-3y`
`=2y^2-2xy+x^2-yxxx^-2-3y`

By implicit differentiation (presumably with respect to x here):
`d/dx y^a=ay^(a-1)xxdy/dx`.
Also using the product rule:
If `f(x) = uv`, then `f'(x)=u'v+uv'`

So the original expression differentiates to:
`4y^1 dy/dx - 2(xdy/dx+yxx1)+2x^1-(-2x^-3y+x^-2dy/dx)-3y^0dy/dx`
`=4ydy/dx-2xdy/dx-2y+2x+(2y)/x^3-1/x^2dy/dx`