Find (dy)/(dx) for x^2=(x−y)/(x+y).

skilpadw3

skilpadw3

Answered question

2022-07-22

Find d y d x for x 2 = x y x + y
I have solved this in two ways.
First, I multiplicated the whole equation by x + y and then I calculated the implicit derivative. I got the following solution:
1 3 x 2 2 x y x 2 + 1
So far so good. When I calculated the implicit derivative of the original expression using the quotient rule though, I got a different solution, i.e.:
x ( y + x ) 2 y x
Can anyone explain to me why I get different solutions ?

Answer & Explanation

Steppkelk

Steppkelk

Beginner2022-07-23Added 11 answers

The two solutions that you found are equal. So where is no contradiction.
Note that solving x 2 = x y x + y for y gives y = x 1 x 2 1 + x 2
Your first solution :
1 3 x 2 2 x y x 2 + 1 = 1 3 x 2 2 x ( x 1 x 2 1 + x 2 ) x 2 + 1 = 1 4 x 2 x 4 ( 1 + x 2 ) 2
Your second solution :
( x + y ) 2 + y x = ( x + x 1 x 2 1 + x 2 ) 2 + 1 x ( x 1 x 2 1 + x 2 ) = 1 4 x 2 x 4 ( 1 + x 2 ) 2
Thus
1 3 x 2 2 x y x 2 + 1 = ( x + y ) 2 + y x

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