Calcule as derivadas parciais partial zpartial x, partial zpartial y, partial^2 zpartial x^2, partial^2 zpartial y^2, e partial^2 zpartial y partial x.
z=x sin ycos (x^2+y^2)

Passo 1

O enunciado pede as derivadas parciais de
z(x,y)=xsin ycosl(x^2+y^2r)
ou seja, vamos usar as regras de derivação para quociente, produto e cadeia. Para simplificar a notação, seja
u = x^2 + y^2, D=cos u, N = xsin y

Passo 2

Para partial zpartial x aplicamos a regra do quociente
partial zpartial x =D,partial Npartial x-N,partial Dpartial xD^2
Aqui
partial Npartial x=sin y, partial Dpartial x =-sin u;partial upartial x =-sin ucdot 2x
Logo
partial zpartial x =cos u;sin y - xsin y;(-2xsin u)cos^2 u =sin yl(cos u+2x^2sin ur)cos^2 u

Passo 3

Para partial zpartial y usamos análogo quociente
partial zpartial y =D,partial Npartial y-N,partial Dpartial yD^2,
onde
partial Npartial y=xcos y, partial Dpartial y =-sin u;partial upartial y =-sin ucdot 2y
Dessa forma
partial zpartial y =cos u;xcos y - xsin y;(-2ysin u)cos^2 u =xl(cos ycos u+2ysin ysin ur)cos^2 u

Passo 4

Para partial^2 zpartial x^2 derivamos partial zpartial x =sin y;A;sec^2u em x, onde A=cos u+2x^2sin u. Usamos produto e cadeia:
partial^2 zpartial x^2 =sin yl(A',x,sec^2u + A,( sec^2u)',xr)
Calcula-se
A',x =2xsin u+4x^3cos u, (sec^2u)',x =4xsec^2utan u
Logo
partial^2 zpartial x^2 =sin y;sec^2ul( 2xsin u+4x^3cos u +4xAtan ur),
com A=cos u+2x^2sin u.

Passo 5

Para partial^2 zpartial y^2 derivamos partial zpartial y =x,B,sec^2u em y, onde B=cos ycos u+2ysin ysin u. Tem-se
partial^2 zpartial y^2 =xl(B',y,sec^2u + B,( sec^2u)',yr),
com
B',y =sin yl(2sin u+(4y^2-1)cos ur), (sec^2u)',y =4ysec^2utan u
Portanto
partial^2 zpartial y^2 =x;sec^2ul( sin yl(2sin u+(4y^2-1)cos ur) +4yBtan ur)

Passo 6

Para partial^2 zpartial ypartial x derivamos partial zpartial x =sin y;A;sec^2u em y. Faz-se
partial^2 zpartial ypartial x =(sin y)',y,A,sec^2u +sin y,A',y,sec^2u +sin y,A,( sec^2u)',y
Com
(sin y)',y=cos y, A',y=2y(-sin u+2x^2cos u), (sec^2u)',y=4ysec^2utan u,
obtém-se
partial^2 zpartial ypartial x =sec^2ul( Acos y +2ysin y(-sin u+2x^2cos u) +4ysin y,Atan u r)

Resposta

partial zpartial x =sin yl(cos u+2x^2sin ur)cos^2u, partial zpartial y =xl(cos ycos u+2ysin ysin ur)cos^2u,
partial^2 zpartial x^2 =sin y;sec^2ul( 2xsin u+4x^3cos u+4x(cos u+2x^2sin u)tan u r),
partial^2 zpartial y^2 =x;sec^2ul( sin yl(2sin u+(4y^2-1)cos ur) +4y(cos ycos u+2ysin ysin u)tan u r),
partial^2 zpartial ypartial x =sec^2ul( (cos u+2x^2sin u)cos y +2ysin y(-sin u+2x^2cos u) +4ysin y(cos u+2x^2sin u)tan u r),
onde u=x^2+y^2.

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