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)=tan ^-1(yx)

Passo 1

O enunciado pede as segundas derivadas parciais de
f(x,y)=tan^-1!l(tfrac yxr),
ou seja, partial^2 fpartial x^2, partial^2 fpartial y^2, partial fpartial ypartial x e partial^2 fpartial xpartial y, e depois verificar que as misturadas são iguais. Vamos usar a regra da cadeia para derivar arctan(u) e a derivada de u=y/x.

Passo 2

Calcular as derivadas de primeira ordem. Seja u=y/x. Então
partial fpartial x =11+u^2,partialpartial x!l(tfrac yxr) =11+(y/x)^2,l(-tfracyx^2r) =-yx^2+y^2,
partial fpartial y =11+u^2,partialpartial y!l(tfrac yxr) =11+(y/x)^2,l(tfrac1xr) =xx^2+y^2

Passo 3

Calcular partial^2 fpartial x^2. Derivando partial f/partial x=-y/(x^2+y^2) em relação a x (com y constante) dá
partial^2 fpartial x^2 =-y;ddxl((x^2+y^2)^-1r) =-y;l(-2xr)(x^2+y^2)^-2 =2xy(x^2+y^2)^2

Passo 4

Calcular partial^2 fpartial y^2. Derivando partial f/partial y=x/(x^2+y^2) em relação a y (com x constante) dá
partial^2 fpartial y^2 =x;ddyl((x^2+y^2)^-1r) =x;l(-2yr)(x^2+y^2)^-2 =-2xy(x^2+y^2)^2

Passo 5

Calcular partial fpartial ypartial x. Derivando partial f/partial y=x/(x^2+y^2) em relação a x
partialpartial x!l(xx^2+y^2r) =(x^2+y^2)-xcdot2x(x^2+y^2)^2 =y^2-x^2(x^2+y^2)^2

Passo 6

Calcular partial^2 fpartial xpartial y. Derivando partial f/partial x=-y/(x^2+y^2) em relação a y
partialpartial y!l(-yx^2+y^2r) =-(x^2+y^2)+ycdot2y(x^2+y^2)^2 =y^2-x^2(x^2+y^2)^2

Passo 7

Verificar que as mistas coincidem:
partial fpartial ypartial x =y^2-x^2(x^2+y^2)^2 =partial^2 fpartial xpartial y

Resposta

(a) partial^2 fpartial x^2=2xy(x^2+y^2)^2
(b) partial^2 fpartial y^2=-2xy(x^2+y^2)^2
(c) partial fpartial ypartial x=y^2-x^2(x^2+y^2)^2
(d) partial^2 fpartial xpartial y=y^2-x^2(x^2+y^2)^2
(e) Conferido que partial fpartial ypartial x=partial^2 fpartial xpartial y

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