There is no trouble when I have to differentiate a function f ( x ,...

equissupnica7

equissupnica7

Answered

2022-07-16

There is no trouble when I have to differentiate a function f ( x , y ) where y = g ( x ) using Implicit Function Theorem. It's easy to get g x x and g y y . For instance, the formula for g x x = f x x F y 2 ( F x y + F y x ) F x F y + F y y F x 2 F y 3 knowing that g ( x ) = F x F y whenever F y 0 .
Now, i'm in the situation z + x + ( y + z ) 4 = 0 where it defines an implicit function z = f ( x , y ).
I got f x and f y using the theorem. However, I can't get f x x and f y y from there. I know I shoud apply the chain rule but I get lost whenever trying to get a "general formula" for them.

Answer & Explanation

sviudes7w

sviudes7w

Expert

2022-07-17Added 12 answers

From F ( x , y , z ( x , y ) ) = 0,
x F = F x + F z z x = 0 z x = F x F z 2 x 2 F = F x x + F x z z x + z x ( F z x + F z z z x ) + F z z x x = 0
Plugging in z x and rearranging, it should be clear how to solve for z x x . A similar method holds for y.

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