(valor: 2,0) Seja f: R^2 R tal que partial fpartial x(x, y)=y, partial fpartial y(x, y)=x. Suponha que x=(s-2) e^t wt^3 e y=1-sw+t+tan ^3(t) raizt w.
(a) Formule a regra da cadeia para calcular partial fpartial t, partial fpartial s e partial fpartial w
(b) Calcule partial fpartial t, partial fpartial s e partial fpartial w.

Passo 1

O enunciado dá uma função f(x,y) com
partial fpartial x(x,y)=y, partial fpartial y(x,y)=x
e define
x = (s-2)e^t wt^3, y = 1-sw+t + tan^3(t),raizt w
O conteúdo aqui é a regra da cadeia para funções de várias variáveis: para cada parâmetro uins,t,w,
partial fpartial u =partial fpartial x,partial xpartial u +partial fpartial y,partial ypartial u

Passo 2

Formulamos explicitamente a regra da cadeia para cada derivada:
partial fpartial t = y,partial xpartial t + x,partial ypartial t,
partial fpartial s = y,partial xpartial s + x,partial ypartial s,
partial fpartial w = y,partial xpartial w + x,partial ypartial w

Passo 3

Calculamos as derivadas parciais de x:
x(s,t,w)=(s-2)e^twt^-3
partial xpartial t =(s-2),ddtl(e^twt^-3r) =(s-2)e^twl(w,t^-3-3,t^-4r) =(s-2)e^twt^-4(w,t-3),
partial xpartial s =e^twt^-3, partial xpartial w =(s-2),e^tw,t^-3,(t) =(s-2)e^twt^-2

Passo 4

Calculamos as derivadas parciais de y:
y(s,t,w)=1-sw+t+tan^3(t)raizt w
Para o primeiro termo 1-sw+t,
partialpartial t!l(tfrac1-sw+tr) = -1-s(w+t)^2, partialpartial s!l(tfrac1-sw+tr) = -1w+t, partialpartial w!l(tfrac1-sw+tr) = -1-s(w+t)^2
Para o segundo termo tan^3(t)raizt w, note C=tan^3(t) e D=raizt w:
dCdt=3tan^2(t)sec^2(t), partial Dpartial t=w2raizt w, partial Dpartial w=t2raizt w
Logo
partial ypartial t = -1-s(w+t)^2 +3tan^2(t)sec^2(t),raizt w +tan^3(t),w2raizt w,
partial ypartial s = -1w+t,
partial ypartial w = -1-s(w+t)^2 +tan^3(t),t2raizt w

Passo 5

Substituindo na regra da cadeia e usando partial fpartial x=y, partial fpartial y=x, obtemos
partial fpartial t = y,(s-2)e^twt^-4(w,t-3) + xgl[ -1-s(w+t)^2 +3tan^2(t)sec^2(t)raizt w +tan^3(t)w2raizt w gr],
partial fpartial s = y,e^twt^-3 + xl(-1w+tr),
partial fpartial w = y,(s-2)e^twt^-2 + xl[ -1-s(w+t)^2 +tan^3(t)t2raizt w r]

Resposta

partial fpartial t = y,(s-2)e^twt^-4(w,t-3) + xl[ -1-s(w+t)^2 +3tan^2(t)sec^2(t)raizt w +tan^3(t)w2raizt w r],
partial fpartial s = y,e^twt^-3 - xw+t,
partial fpartial w = y,(s-2)e^twt^-2 + xl[ -1-s(w+t)^2 +tan^3(t)t2raizt w r]

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