Use a Regra da Cadeia paraa achar partial z / partial s e partial z / partial t.
z=x^2 y^3, x=s cos (t), y=s sen(t)

Passo 1

O enunciado pede para calcular as derivadas parciais partial zpartial s e partial zpartial t usando a Regra da Cadeia, sabendo que
z = x^2 y^3, x = scos(t), y = ssin(t)
Vamos primeiro obter partial zpartial x e partial zpartial y, depois partial xpartial s,partial ypartial s e partial xpartial t,partial ypartial t, e finalmente aplicar
partial zpartial s = partial zpartial xpartial xpartial s + partial zpartial ypartial ypartial s, partial zpartial t = partial zpartial xpartial xpartial t + partial zpartial ypartial ypartial t

Passo 2

Calculemos as derivadas parciais de z em relação a x e y.
partial zpartial x = partialpartial xl(x^2 y^3r) = 2x,y^3, partial zpartial y = partialpartial yl(x^2 y^3r) = 3x^2,y^2

Passo 3

Agora as derivadas parciais de x e y em relação a s:
partial xpartial s = partialpartial sl(scos tr) = cos t, partial ypartial s = partialpartial sl(ssin tr) = sin t

Passo 4

Aplicando a Regra da Cadeia para partial zpartial s:
partial zpartial s = l(2x,y^3r)cos t + l(3x^2,y^2r)sin t = 2x,y^3cos t + 3x^2,y^2sin t
Substituindo x = scos t e y = ssin t:
2,(scos t),(ssin t)^3cos t +3,(scos t)^2,(ssin t)^2sin t =2s^4cos^2t,sin^3t+3s^4cos^2t,sin^3t =5s^4cos^2t,sin^3t

Passo 5

Calculemos as derivadas parciais de x e y em relação a t:
partial xpartial t = partialpartial tl(scos tr) = -ssin t, partial ypartial t = partialpartial tl(ssin tr) = scos t

Passo 6

Aplicando a Regra da Cadeia para partial zpartial t:
partial zpartial t = (2x,y^3)(-ssin t)+(3x^2,y^2)(scos t) = -2x,y^3,ssin t +3x^2,y^2,scos t
Substituindo x = scos t e y = ssin t:
-2,(scos t),(ssin t)^3,ssin t +3,(scos t)^2,(ssin t)^2,scos t =-2s^5cos t,sin^4t+3s^5cos^3t,sin^2t
Fatorando s^5cos t,sin^2t:
partial zpartial t = s^5cos t,sin^2tl(3cos^2t-2sin^2tr)

Resposta

partial zpartial s = 5 s^4 cos^2 t , sin^3 t,
partial zpartial t = s^5 cos t , sin^2 t l(3 cos^2 t - 2 sin^2 tr)

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