Nos casos abaixo, calcule as derivadas parciais partial fpartial x, partial fpartial y e partial fpartial x partial y de 1 ordem:
f(x, y)=raiza^2-x^2-y^2

Passo 1

O enunciado pede as derivadas parciais de primeira ordem partial fpartial x, partial fpartial y e a derivada mista partial^2 fpartial xpartial y da função
f(x,y)=raiza^2-x^2-y^2
O que vamos usar: regra da cadeia para funções compostas e a definição de derivada parcial.

Passo 2

Para calcular partial fpartial x e partial fpartial y, note que
f(x,y) = (a^2-x^2-y^2)^1/2
Defina u=a^2-x^2-y^2, então f=u^1/2. Pela regra da cadeia,
partial fpartial x =12u^-1/2,partial upartial x =12(a^2-x^2-y^2)^-1/2cdot(-2x) =-xraiza^2-x^2-y^2,
partial fpartial y =12u^-1/2,partial upartial y =12(a^2-x^2-y^2)^-1/2cdot(-2y) =-yraiza^2-x^2-y^2

Passo 3

Agora calculamos a derivada mista
partial^2 fpartial xpartial y =partialpartial yl(partial fpartial xr)
Como partial fpartial x=-x,(a^2-x^2-y^2)^-1/2, temos
partial^2 fpartial xpartial y =-xcdotl(-tfrac12r)(a^2-x^2-y^2)^-3/2cdot(-2y) =-xy(a^2-x^2-y^2)^3/2

Resposta

partial fpartial x=-xraiza^2-x^2-y^2,
partial fpartial y=-yraiza^2-x^2-y^2,
partial^2fpartial xpartial y=-xy(a^2-x^2-y^2)^3/2

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