11^circ ) Determine a diferencial total de primeira ordem de:
w=e^x y-4 x z+y z

Passo 1

O enunciado pede a diferencial total de primeira ordem da função
w(x,y,z)=e^xy-4xz+yz
Isso envolve usar a definição de diferencial total:
dw=partial wpartial x,dx+partial wpartial y,dy+partial wpartial z,dz

Passo 2

Vamos calcular cada derivada parcial:
partial wpartial x =partialpartial xl(e^xyr) -partialpartial x(4xz) +partialpartial x(yz) =y,e^xy-4z+0
partial wpartial y =partialpartial yl(e^xyr) -partialpartial y(4xz) +partialpartial y(yz) =x,e^xy-0+z
partial wpartial z =partialpartial zl(e^xyr) -partialpartial z(4xz) +partialpartial z(yz) =0-4x+y

Passo 3

Substituindo essas derivadas na fórmula da diferencial total, obtemos
dw =(y,e^xy-4z),dx +(x,e^xy+z),dy +(y-4x),dz

Resposta

boxeddw=(y,e^xy-4z),dx+(x,e^xy+z),dy+(y-4x),dz

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