Exercício 6 - Determine todas as derivadas parciais de segunda ordem da função:
G(t, s, z)=cos (t s)+sen(z^3).

Passo 1

O enunciado pede as derivadas parciais de segunda ordem da função
G(t,s,z)=cos(ts)+sinl(z^3r)
O conteúdo aqui é diferenciação parcial, usando regra da cadeia e, quando necessário, produto.

Passo 2

Primeiro calculamos as derivadas parciais de primeira ordem por regra da cadeia:
partial Gpartial t =partialpartial tl[cos(ts)r] =-sin(ts),partial (ts)partial t =-ssin(ts)
partial Gpartial s =partialpartial sl[cos(ts)r] =-sin(ts),partial (ts)partial s =-tsin(ts)
partial Gpartial z =partialpartial zl[sin(z^3)r] =cos(z^3),partial (z^3)partial z =3z^2cos(z^3)

Passo 3

Agora as derivadas puras de segunda ordem:
partial^2 Gpartial t^2 =partialpartial tl[-ssin(ts)r] =-scos(ts),partial (ts)partial t =-s^2cos(ts)
partial^2 Gpartial s^2 =partialpartial sl[-tsin(ts)r] =-tcos(ts),partial (ts)partial s =-t^2cos(ts)
partial^2 Gpartial z^2 =partialpartial zl[3z^2cos(z^3)r] =6zcos(z^3) +3z^2l[-sin(z^3)r]cdot3z^2 =6zcos(z^3)-9z^4sin(z^3)

Passo 4

Em seguida as derivadas mistas de segunda ordem. Usando que a parte em z é independente de t,s:
partial^2 Gpartial t,partial s =partialpartial sl[-ssin(ts)r] =-sin(ts)-scos(ts),partial(ts)partial s =-sin(ts)-stcos(ts)
partial^2 Gpartial t,partial z =partialpartial zl[-ssin(ts)r] =0
partial^2 Gpartial s,partial z =partialpartial zl[-tsin(ts)r] =0
Pela igualdade das mistas (teorema de Clairaut), partial^2G/partial spartial t=partial^2G/partial tpartial s, e o mesmo vale entre z e t ou s.

Resposta

partial^2 Gpartial t^2=-s^2cos(ts), partial^2 Gpartial s^2=-t^2cos(ts), partial^2 Gpartial z^2=6zcos(z^3)-9z^4sin(z^3),
partial^2 Gpartial tpartial s =partial^2 Gpartial spartial t =-sin(ts)-stcos(ts), partial^2 Gpartial tpartial z =partial^2 Gpartial zpartial t =0, partial^2 Gpartial spartial z =partial^2 Gpartial zpartial s =0

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