Chain rule of partial derivatives for composite functions. Function of the form f(x^2+y^2) How do I find the partial derivatives df/dy, df/dx

unjulpild9b 2022-09-20 Answered
Chain rule of partial derivatives for composite functions.
Function of the form
f ( x 2 + y 2 )
How do I find the partial derivatives
f y , f x
How f ( x 2 + y 2 ) behaves. Assuming it should of the form
g ( x , y ) 2 y or h ( x , y ) 2 x
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Answers (2)

ticotaku86
Answered 2022-09-21 Author has 12 answers
Since f ( . ) is univariate function we have
f x = 2 x f ( x 2 + y 2 )
similarly
f y = 2 y f ( x 2 + y 2 )
therefore the differential is
d f = 2 x f ( x 2 + y 2 ) d x + 2 y f ( x 2 + y 2 ) d y
therefore
g ( x , y ) = h ( x , y ) = f ( x 2 + y 2 )
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mikioneliir
Answered 2022-09-22 Author has 2 answers
Use some different notation:
f ( x 2 + y 2 ) =: f ( g ( x , y ) ) = ( f g ) ( x , y )
Now the partial derivatives can be computed by the chain rule (in multiple dimensions):
f x = f g g x = f g 2 x .
f y = f g g y = f g 2 y .
Since you don't know anything else about f, you can't simplify these terms further.
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