find derivative of 12 = y^2 + x^2

Question

how do i find the first derivative of 12 = y² + x²?

which rule do i apply?

Answer 1

Will show you TWO (2) ways to do it:

Option #1:

12 = x^2 + y^2

12-x^2 =y^2

sqrt(12-x^2) =y

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

(1/2)(12-x^2)^(-1/2)(-2x) =y'

-x/(12-x^2)^(1/2)=y'

-x/sqrt(12-x^2)=y'

the domain of the derivative is 12-x^2>=0

(sqrt(12)+x)(sqrt(12)-x)>=0

[-sqrt(12), sqrt(12)]

Option 2

implicit differentiation:

0 =2 y y' + 2x

DIvides by 2:

0 = y y' + x

-x = y y'

y' = -x/y

Now, upon solving the original for y:

12 -x^2 =y^2

+or-sqrt(12-x^2)=y

SUbstitutes:

y' = -x/( sqrt(12-x^2))

the domain of the derivative is 12-x^2>=0

(sqrt(12)+x)(sqrt(12)-x)>=0

[-sqrt(12), sqrt(12)]