How to find the derivative of ln(x^2+y^2)?

ofraun4ys5

ofraun4ys5

Answered question

2023-03-17

How to find the derivative of ln ( x 2 + y 2 ) ?

Answer & Explanation

Desiree Cantrell

Desiree Cantrell

Beginner2023-03-18Added 3 answers

Implicit differentiation and partial differentiation are the two possible directions here.
For implicit differentiation, we have both variables ( x and y ) into the derivative at once, while for partial differentiation we work with each one separately.
Implicitly differetiating, then, we must resort to chain rule, by naming u = x 2 + y 2 and, therefore, considering our original funcion z = ln ( u ) . As the chain rule states that:
d z d u d u d ( x y ) = d z d ( x y ) , then
d z d x = ( 1 u ) ( 2 x + 2 y ) = ( 1 x 2 + y 2 ) ( 2 x + 2 y ) = 2 x + 2 y x 2 + y 2
Now, going to partial differentiation: we keep the chain rule logic, but in the end, we proceed differently, differentiating only one of the two variables, as follows:
δ z δ x = ( 1 u ) ( 2 x ) = 2 x x 2 + y 2
δ z δ y = ( 1 u ) ( 2 y ) = 2 y x 2 + y 2

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