Determine partial z / partial x e partial z / partial y sabendo que
y z=ln (x+z)

Passo 1

O enunciado fornece a relação implícita
y,z = ln(x+z)
onde z = z(x,y). O objetivo é aplicar diferenciação implícita para encontrar partial z/partial x e partial z/partial y.

Passo 2

Derivando ambos os lados em relação a x (com y constante), temos:
LHS: partialpartial xl(y,zr) = y,partial zpartial x,
RHS: partialpartial xl[ln(x+z)r] = 1x+z,l(1 + partial zpartial xr)
Logo,
y,partial zpartial x = 1 + partial zpartial xx+z

Passo 3

Isolando partial zpartial x:
y(x+z),partial zpartial x = 1 + partial zpartial x,
l(y(x+z) - 1r),partial zpartial x = 1,
boxedpartial zpartial x = 1y(x+z) - 1

Passo 4

Derivando ambos os lados em relação a y (com x constante), temos:
LHS: partialpartial yl(y,zr) = z + y,partial zpartial y,
RHS: partialpartial yl[ln(x+z)r] = 1x+z,partial zpartial y
Logo,
z + y,partial zpartial y = partial zpartial yx+z

Passo 5

Isolando partial zpartial y:
y,partial zpartial y - 1x+z,partial zpartial y = -,z,
l(y - 1x+zr),partial zpartial y = -,z,
partial zpartial y = -,zy - tfrac1x+z = -,z,(x+z)y(x+z) - 1

Resposta

partial zpartial x = 1y(x+z) - 1, partial zpartial y = -,z,(x+z)y(x+z) - 1

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