19^circ) Derivar 2 x^2 y z-4 x y^2 z^2+6 x z^3-4 y z+1=0 .

Passo 1

O enunciado dá a relação implícita
F(x,y,z)=2x^2y z ;-;4x y^2z^2;+;6x z^3;-;4y z;+;1;=;0
e pede a diferenciação implícita para obter partial zpartial x (mantendo y constante). Usamos a condição
partial Fpartial x ;+;partial Fpartial z,partial zpartial x=0

Passo 2

Calculamos a derivada parcial de F em relação a x, tratando y e z como constantes:
partial Fpartial x = partialpartial xl(2x^2y zr) - partialpartial xl(4x y^2z^2r) + partialpartial xl(6x z^3r) - partialpartial xl(4y zr) + partialpartial x(1)
Como
partialpartial x(2x^2y z)=4x y z, partialpartial x(4x y^2z^2)=4y^2z^2, partialpartial x(6x z^3)=6z^3, partialpartial x(4y z)=0,
ficamos com
partial Fpartial x =4x y z ;-;4y^2z^2;+;6z^3

Passo 3

Agora a derivada parcial de F em relação a z, tratando x e y constantes:
partial Fpartial z = partialpartial z(2x^2y z) - partialpartial z(4x y^2z^2) + partialpartial z(6x z^3) - partialpartial z(4y z) + partialpartial z(1)
Como
partialpartial z(2x^2y z)=2x^2y, partialpartial z(4x y^2z^2)=8x y^2z, partialpartial z(6x z^3)=18x z^2, partialpartial z(4y z)=4y,
obtemos
partial Fpartial z =2x^2y;-;8x y^2z;+;18x z^2;-;4y

Passo 4

Pela fórmula da diferenciação implícita
partial Fpartial x ;+;partial Fpartial z,partial zpartial x=0 Longrightarrow partial zpartial x =- partial Fpartial x partial Fpartial z
Substituindo os valores de partial F/partial x e partial F/partial z, resulta
partial zpartial x =-4x y z ;-;4y^2z^2;+;6z^3 2x^2y;-;8x y^2z;+;18x z^2;-;4y

Resposta

boxed partial zpartial x =-4x y z -4y^2z^2+6z^3 2x^2y -8x y^2z +18x z^2-4y

Exercícios relacionados