I want to solve dy/dx for the following: x^2+y^2=R^2 where R is a constant. I know to use implicit differentiation, though I have a question. When I derive R^2, do I obtain 2R or 0? Additionally, deriving y^2 with respect to x yields 2y(dy/dx)? This is different from a partial derivative?

Adrianna Macias 2022-07-18 Answered
I want to solve d y / d x for the following:
x 2 + y 2 = R 2 where R is a constant.
I know to use implicit differentiation, though I have a question. When I derive R 2 , do I obtain 2 R or 0?
Additionally, deriving y 2 with respect to x yields 2 y ( d y / d x )? This is different from a partial derivative?
Thanks!
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Answers (2)

bulgarum87
Answered 2022-07-19 Author has 16 answers
Suppose R is a function of x and y; then
(1) x 2 + y 2 = R 2 ( x , y ) ;
if we define
(2) F ( x , y ) = x 2 + y 2 R 2 ( x , y ) ,
we may also write (1) as
(3) F ( x , y ) = x 2 + y 2 R 2 ( x , y ) = 0 ;
by the implicit funtion theorem, this equation in fact may be seen as defining y ( x ), a function of x, provided that
(4) F ( x , y ) y 0 ;
we have
(5) F ( x , y ) y = 2 y 2 R ( x , y ) R ( x , y ) y 0
provided
(6) y R ( x , y ) R ( x , y ) y ;
under such circumstances, we may affirm y ( x ) is uniquely determined as a differentiable function of x in some neighborhood of any point ( x , y ); then we have
(7) F ( x , y ) = x 2 + y 2 ( x ) R 2 ( x , y ( x ) ) = 0 ;
we may take the total derivative with respect to x to obtain
(8) d F ( x , y ) d x = 2 x + 2 y d y ( x ) d x 2 R ( x , y ( x ) ) ( R ( x , y ( x ) ) x + R ( x , y ( x ) ) y d y ( x ) d x ) = 0 ;
a little algebra allows us to isolate the terms containing d y ( x ) / d x:
(9) x + y d y ( x ) d x R ( x , y ( x ) ) ( R ( x , y ( x ) ) x + R ( x , y ( x ) ) y d y ( x ) d x ) = 0 ;
(10) y d y ( x ) d x R ( x , y ( x ) ) R ( x , y ( x ) ) y d y ( x ) d x = R ( x , y ( x ) ) R ( x , y ( x ) ) x x ;
(11) ( y R ( x , y ( x ) ) R ( x , y ( x ) y ) d y ( x ) d x = R ( x , y ( x ) ) R ( x , y ( x ) ) x x ;
for the sake of compactess and brevity, we introduce the subscript notation
(12) R x = R x , etc. ,
and write (11) in the form
(13) y ( x ) = R R x x y R R y = x R R x y R R y ,
which gives a general expression for y ( x ); in the event that R ( x , y ) is constant, we obtain
(14) y ( x ) = x y ,
which the reader may recognize as the slope of the circle
(15) x 2 + y 2 = R 2
at any point ( x , y ) where y 0.
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Freddy Friedman
Answered 2022-07-20 Author has 5 answers
By the chaine rule you will get
2 x + 2 y y = 0
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