`f(x,y)=x^2+xy+y^2=9`
Find Implicit differentiation, Derivative with respect to x

Solution:
Method-1: using Chain Rule
`x^2+y^2+xy-9=0`

After implicit differentiation

`d/(dx)(x^2+y^2+xy-9)=2x+2yy'+y+xy'`

`d/(dx)(x^2+y^2+xy-9)`

`=d/(dx)(x^2)+d/(dx)(y^2)+d/(dx)(xy)-d/(dx)(9)`

`d/(dx)(y^2)=2yy'`

`d/(dx)(y^2)`

`=2y*d/(dx)(y)`

`=2y*y'`

`=2yy'`

`d/(dx)(xy)=y+xy'`

`d/(dx)(xy)`

`=(d/(dx)(x))y+x(d/(dx)(y))`

`=1y+x(y')`

`=y+xy'`

`=2x+2yy'+(y+xy')-0`

`=2x+2yy'+y+xy'-0`

`=2x+2yy'+y+xy'`

`2x+2yy'+y+xy'=0`

`=>2yy'+xy'+2x+y = 0`

`=>2yy'+xy' = -2x-y`

`=>y'(2y+x) = -2x-y`

`=>y' = (-2x-y)/(2y+x)`

`=>y' = ((-2x-y)/(2y+x))`

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Method-2: using partial derivatives `Fx,Fy`. If `F(x,y)=0`, then `y'=-(Fx)/(Fy)`

`Fx=(del)/(del x)(x^2+y^2+xy-9)=2x+y`

Now,
`del/(delx)(x^2+y^2+xy-9)`

`=del/(delx)(x^2)+del/(delx)(y^2)+del/(delx)(xy)-del/(delx)(9)`

`del/(delx)(y^2)=0`

`del/(delx)(y^2)`

`=2y*del/(delx)(y)`

`=2y*0`

`=0`

`del/(delx)(xy)=y+0`

`del/(delx)(xy)`

`=(del/(delx)(x))y+x(del/(delx)(y))`

`=1y+x*0`

`=y+0`

`=2x+0+(y+0)-0`

`=2x+0+y+0-0`

`=2x+y`

`:.(del)/(del x)(x^2+y^2+xy-9)=2x+y`

`Fy=(del)/(del y)(x^2+y^2+xy-9)=2y+x`

Now,
`del/(dely)(x^2+y^2+xy-9)`

`=del/(dely)(x^2)+del/(dely)(y^2)+del/(dely)(xy)-del/(dely)(9)`

`del/(dely)(x^2)=0`

`del/(dely)(x^2)`

`=2x*del/(dely)(x)`

`=2x*0`

`=0`

`del/(dely)(xy)=0+x`

`del/(dely)(xy)`

`=(del/(dely)(x))y+x(del/(dely)(y))`

`=0y+x*1`

`=0+x`

`=0+2y+(0+x)-0`

`=0+2y+0+x-0`

`=2y+x`

`:.(del)/(del y)(x^2+y^2+xy-9)=2y+x`

`y'=-(Fx)/(Fy)`

`:.y'=-(2x+y)/(2y+x)`

`:.y'=(-(2x+y))/(2y+x)`

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