Implicit differentiation Example x^2+y^2=15, derivative with respect to x

`f(x,y)=x^2+y^2=15`
Find Implicit differentiation, Derivative with respect to x

Solution:
Method-1: using Chain Rule
`x^2+y^2-15=0`

After implicit differentiation

`d/(dx)(x^2+y^2-15)=2x+2yy'`

`d/(dx)(x^2+y^2-15)`

`=d/(dx)(x^2)+d/(dx)(y^2)-d/(dx)(15)`

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

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

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

`=2y*y'`

`=2yy'`

`=2x+2yy'-0`

`=2x+2yy'`

`2x+2yy'=0`

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

`=>2yy' = -2x`

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

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

`=>y' = -x/y`

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

`Fx=(del)/(del x)(x^2+y^2-15)=2x`

Now,
`del/(delx)(x^2+y^2-15)`

`=del/(delx)(x^2)+del/(delx)(y^2)-del/(delx)(15)`

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

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

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

`=2y*0`

`=0`

`=2x+0-0`

`=2x`

`:.(del)/(del x)(x^2+y^2-15)=2x`

`Fy=(del)/(del y)(x^2+y^2-15)=2y`

Now,
`del/(dely)(x^2+y^2-15)`

`=del/(dely)(x^2)+del/(dely)(y^2)-del/(dely)(15)`

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

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

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

`=2x*0`

`=0`

`=0+2y-0`

`=2y`

`:.(del)/(del y)(x^2+y^2-15)=2y`

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

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

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

This material is intended as a summary. Use your textbook for detail explanation.
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1. Example `x^2+y^2=15`, derivative with respect to x