How to find y'' by implicit differentiation for 4x^2+3y^2=6?

Evelyn Buchanan

Evelyn Buchanan

Answered question

2023-03-01

How to find y'' by implicit differentiation for 4 x 2 + 3 y 2 = 6 ?

Answer & Explanation

Giovanna Shaw

Giovanna Shaw

Beginner2023-03-02Added 3 answers

4 x 2 + 3 y 2 = 6
d d x ( 4 x 2 + 3 y 2 ) = d d x ( 6 )
8 x + 6 y d y d x = 0
So d y d x = - 4 x 3 y
The second derivative follows:
d d x ( d y d x ) = d d x ( - 4 x 3 y )
d 2 y d x 2 = ( - 4 ) ( 3 y ) - ( - 4 x ) ( 3 d y d x ) ( 3 y ) 2
d 2 y d x 2 = - 12 y + 12 x d y d x 9 y 2
Replacing d y d x by the expression above gives us:
d 2 y d x 2 = - 12 y + 12 x ( - 4 x 3 y ) 9 y 2 = - 12 y + - 16 x 2 y 9 y 2
No multiply by y y to get
d 2 y d x 2 = - 12 y 2 - 16 x 2 9 y 3 = - 4 ( 4 x 2 + 3 y 2 ) 9 y 3
Now, use the initial equation: 4 x 2 + 3 y 2 = 6 to write the answer as:
d 2 y d x 2 = = - 24 9 y 3 = - 8 3 y 3

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