1. (1,0 ponto) Assuma que
z=z(x, y)>0
Determine partial zpartial x e partial zpartial y, sabendo que
y z+x ln z=z^2

Passo 1

O enunciado dá a relação implícita
y,z + x ln z = z^2,
onde z = z(x,y)>0. O conteúdo é diferenciação implícita para encontrar partial zpartial x e partial zpartial y. Vamos usar a função
F(x,y,z) = y,z + x ln z - z^2
e aplicar partial Fpartial x + partial Fpartial z,partial zpartial x=0 e partial Fpartial y + partial Fpartial z,partial zpartial y=0.

Passo 2

Calcular as derivadas parciais de F. Primeiro
partial Fpartial x = ln z, partial Fpartial y = z,
e
partial Fpartial z = y + x1z - 2z

Passo 3

Para partial zpartial x usamos
partial Fpartial x + partial Fpartial z,partial zpartial x=0 ;Longrightarrow; ln z + l(y + xz -2zr)partial zpartial x=0
Assim
partial zpartial x = -ln z,y + xz -2z, = -z,ln z,yz + x -2z^2,

Passo 4

Para partial zpartial y usamos
partial Fpartial y + partial Fpartial z,partial zpartial y=0 ;Longrightarrow; z + l(y + xz -2zr)partial zpartial y=0
Logo
partial zpartial y = -z,y + xz -2z, = -z^2,yz + x -2z^2,

Resposta

partial zpartial x = -z,ln z,yz + x -2z^2,, partial zpartial y = -z^2,yz + x -2z^2,

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