Question

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

Answer 1

`y'=-(16xsqrt(2x+xy)+2+y)/(x+2sqrt(2x+xy))`

Explanation:

Assuming that `y=y(x)` we get by the chain rule

`8x+1/2(2x+xy)^(-1/2)(2+y+xy')+y'=0`
multiplying by `2sqrt(2x+xy)` we get

`16xsqrt(2x+xy)+2+y+xy'+2y'sqrt(2x+xy)=0` isolating `y'`

`16xsqrt(2x+xy)+2+y+y'(x+2sqrt(2x+xy))=0` so we get

`y'=-(16xsqrt(2x+xy)+2+y)/(x+2sqrt(2x+xy))`