Calcular as derivadas parciais de 2^ a ordem das funções abaixo:
1.1. z=x^2-3 y^3+4 x^2 y^2
1.2. z=x^2 y^2-x y
1.3. z=ln x y
1.4. z=e^x y

Passo 1

O enunciado pede para calcular as derivadas parciais de segunda ordem das funções
1.1 z=x^2-3y^3+4x^2y^2
1.2 z=x^2y^2-xy
1.3 z=ln(xy)
1.4 z=e^xy
O que vamos usar: para cada função calculamos primeiro partial zpartial x e partial zpartial y aplicando regra do poder, produto e cadeia, e depois derivamos novamente para obter
partial^2zpartial x^2, partial^2zpartial xpartial y e partial^2zpartial y^2.

Passo 2

Para z=x^2-3y^3+4x^2y^2:
partial zpartial x=2x+8xy^2,
partial zpartial y=-9y^2+8x^2y
Derivando de novo:
partial^2zpartial x^2=2+8y^2,
partial^2zpartial xpartial y =partialpartial y(2x+8xy^2)=16xy,
partial^2zpartial y^2 =partialpartial y(-9y^2+8x^2y)=-18y+8x^2

Passo 3

Para z=x^2y^2-xy:
partial zpartial x=2xy^2-y,
partial zpartial y=2x^2y-x
Então
partial^2zpartial x^2=partialpartial x(2xy^2-y)=2y^2,
partial^2zpartial xpartial y =partialpartial y(2xy^2-y)=4xy-1,
partial^2zpartial y^2=partialpartial y(2x^2y-x)=2x^2

Passo 4

Para z=ln(xy) notamos que z=ln x+ln y, então
partial zpartial x=1x, partial zpartial y=1y
Derivando de novo:
partial^2zpartial x^2=-1x^2, partial^2zpartial xpartial y =partialpartial yl(1xr)=0, partial^2zpartial y^2=-1y^2

Passo 5

Para z=e^xy:
partial zpartial x=y,e^xy, partial zpartial y=x,e^xy
Em seguida
partial^2zpartial x^2 =partialpartial x(y,e^xy)=y^2e^xy,
partial^2zpartial xpartial y =partialpartial y(y,e^xy) =e^xy+xy,e^xy=(1+xy)e^xy,
partial^2zpartial y^2 =partialpartial y(x,e^xy)=x^2e^xy

Resposta

1.1
zxx=2+8y^2, zxy=16xy, zyy=-18y+8x^2
1.2
zxx=2y^2, zxy=4xy-1, zyy=2x^2
1.3
zxx=-1x^2, zxy=0, zyy=-1y^2
1.4
zxx=y^2e^xy, zxy=(1+xy)e^xy, zyy=x^2e^xy

Exercícios relacionados