Calcule as derivadas parciais partial zpartial x, partial zpartial y, partial^2 zpartial x^2, partial^2 zpartial y^2, e partial^2 zpartial y partial x.
z=arcsin (x^2+y)

Passo 1

O enunciado define
z=arcsinl(x^2+yr)
e pede todas as derivadas parciais de primeira e segunda ordem. Vamos usar a regra da cadeia para funções compostas e a fórmula
dduarcsin(u)=1raiz1-u^2,

Passo 2

Para partial zpartial x defina T=x^2+y. Então
partial Tpartial x=2x e aplicando a cadeia:
partial zpartial x =1raiz1-T^2,partial Tpartial x =2xraiz1-(x^2+y)^2,

Passo 3

Para partial zpartial y usa-se partial Tpartial y=1. Logo
partial zpartial y =1raiz1-T^2 =1raiz1-(x^2+y)^2,

Passo 4

Agora partial^2zpartial x^2 =partialpartial x!gl(2xraiz1-T^2gr).
Usando produto e cadeia, onde
partialpartial x(2x)=2 e
partialpartial x(1-T^2)^-1/2 =2x,T,(1-T^2)^-3/2. Portanto,
partial^2zpartial x^2 =2(1-T^2)^-1/2+2xcdot 2xT(1-T^2)^-3/2 =2(1-(x^2+y)^2)^-1/2 +4x^2(x^2+y)(1-(x^2+y)^2)^-3/2,

Passo 5

Para partial^2zpartial y^2 =partialpartial y!l((1-T^2)^-1/2r), a cadeia dá
partialpartial y(1-T^2)^-1/2 =T(1-T^2)^-3/2. Logo
partial^2zpartial y^2 =(x^2+y),(1-(x^2+y)^2)^-3/2,

Passo 6

Por fim,
partial^2zpartial ypartial x =partialpartial y!l(2xraiz1-T^2r).
Aqui 2x é constante em y e
partialpartial y(1-T^2)^-1/2 =T(1-T^2)^-3/2. Assim
partial^2zpartial ypartial x =2x,(x^2+y),(1-(x^2+y)^2)^-3/2,

Resposta

partial zpartial x=2xraiz1-(x^2+y)^2
partial zpartial y=1raiz1-(x^2+y)^2
partial^2zpartial x^2 =2(1-(x^2+y)^2)^-1/2 +4x^2(x^2+y)(1-(x^2+y)^2)^-3/2
partial^2zpartial y^2 =(x^2+y)(1-(x^2+y)^2)^-3/2
partial^2zpartial ypartial x =2x,(x^2+y),(1-(x^2+y)^2)^-3/2

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