Nos problemas de 1 a 12 (a) calcule partial^2 fpartial x^2 , (b) calcule partial^2 fpartial y^2 , (c) calcule
partial fpartial y partial x , (d) calcule partial^2 fpartial x partial y , e (e) verifique que partial^2 fpartial y partial x=partial^2 fpartial x partial y .
f(x, y)=x+yx-y

Passo 1

O enunciado pede as segundas derivadas parciais de
f(x,y)=x+yx-y
isto é, vamos usar a regra do quociente para encontrar partial fpartial x e partial fpartial y e depois derivar novamente para obter partial^2 fpartial x^2, partial^2 fpartial y^2, partial^2 fpartial ypartial x e partial^2 fpartial xpartial y. No final, conferimos a igualdade dos mistos.

Passo 2

Calcular partial fpartial x usando
partialpartial xl(x+yx-yr) =(1)(x-y)-(x+y)(1)(x-y)^2 =x-y-x-y(x-y)^2 =-2y(x-y)^2

Passo 3

Calcular partial^2 fpartial x^2 derivando partial fpartial x=-2y,(x-y)^-2 em relação a x:
partial^2 fpartial x^2 =-2ycdot(-2),(x-y)^-3cdot1 =4y(x-y)^3

Passo 4

Calcular partial fpartial y usando
partialpartial yl(x+yx-yr) =(1)(x-y)-(x+y)(-1)(x-y)^2 =x-y+x+y(x-y)^2 =2x(x-y)^2

Passo 5

Calcular partial^2 fpartial y^2 derivando partial fpartial y=2x,(x-y)^-2 em relação a y:
partial^2 fpartial y^2 =2xcdot(-2),(x-y)^-3cdot(-1) =4x(x-y)^3

Passo 6

Calcular partial^2 fpartial ypartial x derivando partial fpartial x=-2y,(x-y)^-2 em relação a y:
partial^2 fpartial ypartial x =-2,(x-y)^-2 -2ycdot(-2),(x-y)^-3cdot(-1) =-2(x-y)-4y(x-y)^3 =-2x-2y(x-y)^3 =-2,x+y(x-y)^3

Passo 7

Calcular partial^2 fpartial xpartial y derivando partial fpartial y=2x,(x-y)^-2 em relação a x:
partial^2 fpartial xpartial y =2,(x-y)^-2 +2xcdot(-2),(x-y)^-3cdot1 =2(x-y)-4x(x-y)^3 =-2x-2y(x-y)^3 =-2,x+y(x-y)^3

Passo 8

Comparando os mistos:
partial^2 fpartial ypartial x =-2,x+y(x-y)^3 e partial^2 fpartial xpartial y =-2,x+y(x-y)^3,
logo são iguais.

Resposta

(a) partial^2 fpartial x^2=4y(x-y)^3
(b) partial^2 fpartial y^2=4x(x-y)^3
(c) partial^2 fpartial ypartial x=-2,x+y(x-y)^3
(d) partial^2 fpartial xpartial y=-2,x+y(x-y)^3
(e) De fato partial^2 fpartial ypartial x=partial^2 fpartial xpartial y

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