Dado z(x, y)=x^2+y^2-4, x(t)=t^2, y(t)=t^3, calcule zt
Dado z(x, y)=2 x^3+3 y^2-1, x(t, s)=s t^2, y(t, s)=t s^3, calcule zt e zs
Dado z=x^2+y^2+x y, x(t)=sen(t), y(t)=e^t, calcule zt
Dado z=x^2 y^3, x(t, s)=s cos (t), y(t, s)=s sen(t), calcule zt e zs

Passo 1

O enunciado apresenta quatro composições de funções z(x,y) com variáveis x(t), y(t) ou x(t,s), y(t,s) e pede derivadas como zt e zs. Para isso vamos usar a regra da cadeia, que em geral diz:
dzdt = partial zpartial x,dxdt + partial zpartial y,dydt,
e, quando há duas variáveis livres t e s,
partial zpartial t = partial zpartial x,partial xpartial t + partial zpartial y,partial ypartial t, partial zpartial s = partial zpartial x,partial xpartial s + partial zpartial y,partial ypartial s

Passo 2

Para z(x,y)=x^2+y^2-4 com x(t)=t^2 e y(t)=t^3:
partial zpartial x=2x, partial zpartial y=2y,
x'=2t, y'=3t^2.
Logo,
zt =2xcdot2t+2ycdot3t^2 =4t,x+6t^2,y =4tcdot t^2+6t^2cdot t^3 =4t^3+6t^5 =2t^3(2+3t^2)

Passo 3

Para z(x,y)=2x^3+3y^2-1 com x(t,s)=s,t^2 e y(t,s)=t,s^3:
partial zpartial x=6x^2, partial zpartial y=6y.
Temos
xt=2t,s, yt=s^3,
xs=t^2, ys=3t,s^2.
Então
zt =6x^2cdot(2t,s)+6ycdot s^3 =12s,t,x^2+6s^3y =12s,t,(s t^2)^2+6s^3(t,s^3) =12s^3t^5+6s^6t,
zs =6x^2cdot t^2+6ycdot(3t,s^2) =6t^2x^2+18t,s^2y =6t^2(s t^2)^2+18t,s^2(t,s^3) =6s^2t^6+18s^5t^2

Passo 4

Para z(x,y)=x^2+y^2+xy com x(t)=sin t e y(t)=e^t:
partial zpartial x=2x+y, partial zpartial y=2y+x,
x'=cos t, y'=e^t.
Portanto
zt =(2x+y)cos t+(2y+x)e^t =(2sin t+e^t)cos t+(2e^t+sin t)e^t

Passo 5

Para z(x,y)=x^2y^3 com x(t,s)=scos t e y(t,s)=ssin t:
partial zpartial x=2x,y^3, partial zpartial y=3x^2y^2,
xt=-ssin t, yt=scos t,
xs=cos t, ys=sin t.
Logo
zt =2x,y^3(-ssin t)+3x^2y^2(scos t) =-2s,x,y^3sin t+3s,x^2y^2cos t
zt =-2s(scos t)(ssin t)^3sin t +3s(scos t)^2(ssin t)^2cos t
zt =-2s^5cos t,sin^4t +3s^5cos^3t,sin^2t,
zs =2x,y^3cos t+3x^2y^2sin t =2(scos t)(ssin t)^3cos t +3(scos t)^2(ssin t)^2sin t
zs =2s^4cos^2t,sin^3t +3s^4cos^2t,sin^3t =5s^4cos^2t,sin^3t

Resposta

( 1) zt=2t^3(2+3t^2)\n( 2) zt=12s^3t^5+6s^6t, zs=6s^2t^6+18s^5t^2\n( 3) zt=(2sin t+e^t)cos t+(2e^t+sin t)e^t\n( 4) zt=-2s^5cos t,sin^4t+3s^5cos^3t,sin^2t, zs=5s^4cos^2t,sin^3t

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