Usando a regra da cadeia para z=f(x, y) e w=f(x, y, z), calcule:
partial zpartial t, partial zpartial s e partial wpartial t, partial wpartial s e partial wpartial r
z=x^2+y^2, x=cosh (s) cos (t), y=senh(s) sen(t)

Passo 1

O enunciado nos dá a função
z = x^2 + y^2
com
x = cosh(s),cos(t), y = sinh(s),sin(t),
e pede partial z/partial t e partial z/partial s. Em seguida, para
w = fl(x,y,zr),
queremos partial w/partial t, partial w/partial s e partial w/partial r. Vamos usar a regra da cadeia em cada caso.

Passo 2

Primeiro obtemos as derivadas parciais de x e y em relação a t e a s.
partial xpartial t = cosh(s),l(-sin(t)r) = -cosh(s),sin(t), partial xpartial s = sinh(s),cos(t),
partial ypartial t = sinh(s),cos(t), partial ypartial s = cosh(s),sin(t)

Passo 3

Agora aplicamos a regra da cadeia para
z = x^2 + y^2
Como partial z/partial x = 2x e partial z/partial y = 2y, temos
partial zpartial t = partial zpartial x,partial xpartial t + partial zpartial y,partial ypartial t = 2xl(-cosh(s)sin(t)r) + 2yl(sinh(s)cos(t)r),
partial zpartial s = partial zpartial x,partial xpartial s + partial zpartial y,partial ypartial s = 2xl(sinh(s)cos(t)r) + 2yl(cosh(s)sin(t)r)

Passo 4

Para
w = fl(x,y,zr)
usamos a regra da cadeia em três variáveis:
partial wpartial t = partial wpartial x,partial xpartial t + partial wpartial y,partial ypartial t + partial wpartial z,partial zpartial t,
partial wpartial s = partial wpartial x,partial xpartial s + partial wpartial y,partial ypartial s + partial wpartial z,partial zpartial s,
partial wpartial r = partial wpartial x,partial xpartial r + partial wpartial y,partial ypartial r + partial wpartial z,partial zpartial r
Como x e y (e, portanto, z) não dependem de r, temos partial x/partial r=partial y/partial r=partial z/partial r=0, logo
partial wpartial r = 0

Resposta

partial zpartial t = -2x,cosh(s)sin(t) + 2y,sinh(s)cos(t), partial zpartial s = 2x,sinh(s)cos(t) + 2y,cosh(s)sin(t),
partial wpartial t = wxl(-cosh(s)sin(t)r) + wyl(sinh(s)cos(t)r) + wzl(-2x,cosh(s)sin(t)+2y,sinh(s)cos(t)r),
partial wpartial s = wxl(sinh(s)cos(t)r) + wyl(cosh(s)sin(t)r) + wzl(2x,sinh(s)cos(t)+2y,cosh(s)sin(t)r),
partial wpartial r = 0

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