Encontre a derivada parcial indicada usando a regra da cadeia:
u=x e^-y ; x=t g^-1(r s t) ; y=ln (3 r s+5 s t) ; partial upartial r ; partial upartial s ; partial upartial t

Passo 1

O enunciado define
u = x,e^-y,
com
x = t,g^-1(r s t)
e
y = lnl(3rs + 5str),
e pede partial u/partial r, partial u/partial s e partial u/partial t.
Para isso vamos usar a regra da cadeia na forma
partial upartial r = ux,xr + uy,yr, partial upartial s = ux,xs + uy,ys, partial upartial t = ux,xt + uy,yt,
logo precisamos de ux,uy, de xr,xs,xt e de yr,ys,yt.

Passo 2

Calcular as derivadas parciais de u em relação a x e y.
Como
u(x,y)=x,e^-y,
temos, usando derivada de produto e cadeia,
ux = e^-y, uy = xcdotddy(e^-y) = -x,e^-y

Passo 3

Calcular xr,xs,xt a partir de
x = t,g^-1(r s t)
Defina w = r s t. Então x = t,g^-1(w) e
partial wpartial r = s t, partial wpartial s = r t, partial wpartial t = r s
Usando produto e cadeia,
xr = t,(g^-1)'(w),partial wpartial r = t,(g^-1)'(r s t),(s t) = s,t^2,(g^-1)'(r s t),
xs = t,(g^-1)'(r s t),(r t) = r,t^2,(g^-1)'(r s t),
xt = g^-1(r s t) + t,(g^-1)'(r s t),(r s) = g^-1(r s t) + r s t,(g^-1)'(r s t)

Passo 4

Calcular yr,ys,yt a partir de
y = lnl(3rs + 5str)
Pela derivada de logaritmo,
yr = 13 r s + 5 s t,partialpartial r(3rs + 5st) = 3s3rs+5st,
ys = 13 r s + 5 s t,partialpartial s(3rs + 5st) = 3r+5t3rs+5st,
yt = 13 r s + 5 s t,partialpartial t(3rs + 5st) = 5s3rs+5st

Passo 5

Agora aplicamos a regra da cadeia em cada variável:
Para r:
partial upartial r = ux,xr + uy,yr = e^-y,(s,t^2,(g^-1)'(r s t)) ;+;(-x,e^-y);3s3rs+5st
partial upartial r = e^-yl(s,t^2,(g^-1)'(r s t) - x,3s3rs+5str)
Para s:
partial upartial s = e^-y,(r,t^2,(g^-1)'(r s t)) +(-x,e^-y);3r+5t3rs+5st
partial upartial s = e^-yl(r,t^2,(g^-1)'(r s t) - x,3r+5t3rs+5str)
Para t:
partial upartial t = e^-y,l(g^-1(r s t) + r s t,(g^-1)'(r s t)r) +(-x,e^-y);5s3rs+5st
partial upartial t = e^-yl(g^-1(r s t) + r s t,(g^-1)'(r s t) - x,5s3rs+5str)

Resposta

partial upartial r = e^-yl(s,t^2,(g^-1)'(r s t) - x,3s3rs+5str),
partial upartial s = e^-yl(r,t^2,(g^-1)'(r s t) - x,3r+5t3rs+5str),
partial upartial t = e^-yl(g^-1(r s t) + r s t,(g^-1)'(r s t) - x,5s3rs+5str)

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