I am trying to solve the problem: x^2+xy+y^3=0 using implicit differentiation.

joyoshibb 2022-08-14 Answered
I am trying to solve the problem: x 2 + x y + y 3 = 0 using implicit differentiation.
My workings:
( 1 ) d d x [ x 2 ] + d d x [ x y ] + d d x [ y 3 ] = d d x [ 0 ]
( 2 ) 2 x + y + d y 3 d y d y d x = 0
( 3 ) 2 x + y + 3 y 2 ( d y d x ) = 0
( 4 ) d y d x = 2 x + y 3 y 2
But the answer says it should be:
( 3 ) 2 x + y + d x y d y d y d x + 3 y 2 ( d y d x ) = 0
( 4 ) 2 x + y + d y d x ( x + 3 y 2 ) = 0
( 5 ) d y d x = 2 x + y x + 3 y 2
Why?
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Answers (2)

Barbara Klein
Answered 2022-08-15 Author has 19 answers
You can't conclude that d d x ( x y ) = y, since this is effectively the product of the functions x and y. Hence, using the product rule gives
d d x ( x y ) = y d d x ( x ) + x d d x ( y ) = y + x d y d x
The rest should work out as expected.
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Taliyah Reyes
Answered 2022-08-16 Author has 6 answers
Differentials will make the process much more intuitive.
0 = d ( x 2 + x y + y 3 ) 0 = 2 x d x + ( x d y + y d x ) + 3 y 2 d y ( x + 3 y 2 ) d y = ( 2 x + y ) d x d y d x = ( 2 x + y x + 3 y 2 )
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