Use a Regra da Cadeia para achar partial z / partial s e partial z / partial t.
z=e^x+2 y, quad x=s / t, quad y=t / s

Passo 1

O enunciado pede para usar a Regra da Cadeia para achar partial z/partial s e partial z/partial t em z=e^x+2y, com x=s/t e y=t/s. O conteúdo é derivação composta (Regra da Cadeia) e vamos precisar de partial z/partial x, partial z/partial y, partial x/partial s, partial x/partial t, partial y/partial s e partial y/partial t.

Passo 2

Primeiro, derivamos z=e^u com u=x+2y. Temos
partial zpartial x=e^x+2y, partial zpartial y=2e^x+2y

Passo 3

Agora calculamos as derivadas parciais de x e y:
x=st;Rightarrow;partial xpartial s=1t,;partial xpartial t=-st^2;
y=ts;Rightarrow;partial ypartial s=-ts^2,;partial ypartial t=1s

Passo 4

Usando a Regra da Cadeia para partial z/partial s:
partial zpartial s=partial zpartial xpartial xpartial s+partial zpartial ypartial ypartial s=e^x+2yl(1t-2ts^2r)

Passo 5

E para partial z/partial t:
partial zpartial t=partial zpartial xpartial xpartial t+partial zpartial ypartial ypartial t=e^x+2yl(-st^2+2sr)

Passo 6

Finalmente, substituímos x=s/t e y=t/s em e^x+2y=e^s/t+2t/s.

Resposta

partial zpartial s=e^st+2tsl(1t-2ts^2r), partial zpartial t=e^st+2tsl(-st^2+2sr)

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