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=e^sin (x raizy)

Passo 1

O enunciado pede para calcular as derivadas parciais de
z = e^sinl(xraizyr)
em relação a x e y, e em seguida as segundas derivadas partial^2 z/partial x^2, partial^2 z/partial y^2 e a mista partial^2 z/(partial y,partial x). Para isso, vamos usar a regra da cadeia e, quando necessário, a regra do produto.

Passo 2

Calcular partial z/partial x. Seja
u = sinl(xraizyr),
então z = e^u e
partial zpartial x = e^upartial upartial x = e^sin(xraizy)cdotcosl(xraizyr)cdotraizy

Passo 3

Calcular partial z/partial y. Ainda com u=sin(xraizy), temos
partial upartial y = cosl(xraizyr)cdotx2raizy,
logo
partial zpartial y = e^upartial upartial y = e^sin(xraizy)cosl(xraizyr)x2raizy

Passo 4

Calcular partial^2 z/partial x^2. Partindo de
partial zpartial x =raizy,e^sin(xraizy)cos(xraizy),
aplicamos produto e cadeia para derivar em x. O resultado é
partial^2 zpartial x^2 = e^sin(xraizy),yl(cos^2(xraizy)-sin(xraizy)r)

Passo 5

Calcular partial^2 z/partial y^2. Partindo de
partial zpartial y = x2raizy,e^sin(xraizy)cos(xraizy),
aplicamos produto e cadeia em y. O resultado é
partial^2 zpartial y^2 = e^sin(xraizy)l[ -x4,y^3/2cos(xraizy) +x^24,yl(cos^2(xraizy)-sin(xraizy)r) r]

Passo 6

Calcular partial^2 zpartial y,partial x. Derivando
partial zpartial x =raizy,e^sin(xraizy)cos(xraizy)
em relação a y, obtemos
partial^2 zpartial y,partial x = e^sin(xraizy)l[ cos(xraizy)2raizy +x2l(cos^2(xraizy)-sin(xraizy)r) r]

Resposta

partial zpartial x = e^sin(xraizy)cosl(xraizyr)raizy,
partial zpartial y = e^sin(xraizy)cosl(xraizyr)x2raizy,
partial^2 zpartial x^2 = e^sin(xraizy),yl(cos^2(xraizy)-sin(xraizy)r),
partial^2 zpartial y^2 = e^sin(xraizy)l[ -x4,y^3/2cos(xraizy) +x^24,yl(cos^2(xraizy)-sin(xraizy)r) r],
partial^2 zpartial y,partial x = e^sin(xraizy)l[ cos(xraizy)2raizy +x2l(cos^2(xraizy)-sin(xraizy)r) r]

Exercícios relacionados