(Valor 2) Dada a função f(x, y, z)=\ln \left(x^{4}+y^{4}+z^{4}\right)+\cos \left(3 x^{2} y-2 x y^{2}-z^{2}\right), calcule as derivadas parciais de primeira ordem dessa função.

Passo 1

O enunciado dá a função
f(x,y,z)=\ln\bigl(x^{4}+y^{4}+z^{4}\bigr)+\cos\bigl(3x^{2}y-2xy^{2}-z^{2}\bigr)
e pede as derivadas parciais de primeira ordem. Para isso, vamos usar a regra da cadeia em cada termo:
  1. Para \ln(u), temos \partial/\partial w[\ln(u)]=u'/u.
  2. Para \cos(v), temos \partial/\partial w[\cos(v)]=-\sin(v)\,v'.

Passo 2

Cálculo de \displaystyle \frac{\partial f}{\partial x}:
  • No termo \ln(x^{4}+y^{4}+z^{4}), u=x^{4}+y^{4}+z^{4} e \partial u/\partial x=4x^{3}, então
\frac{\partial}{\partial x}\ln(u) =\frac{4x^{3}}{x^{4}+y^{4}+z^{4}}.
  • No termo \cos(3x^{2}y-2xy^{2}-z^{2}), v=3x^{2}y-2xy^{2}-z^{2} e \partial v/\partial x=6xy-2y^{2}, então
\frac{\partial}{\partial x}\cos(v) =-\sin\bigl(v\bigr)\,(6xy-2y^{2}).
Combinando, resulta
\frac{\partial f}{\partial x} =\frac{4x^{3}}{x^{4}+y^{4}+z^{4}} -\sin\bigl(3x^{2}y-2xy^{2}-z^{2}\bigr)\,(6xy-2y^{2}).

Passo 3

Cálculo de \displaystyle \frac{\partial f}{\partial y}:
  • Em \ln(x^{4}+y^{4}+z^{4}), \partial/\partial y \tfrac{4y^{3}}{x^{4}+y^{4}+z^{4}}.
  • Em \cos(3x^{2}y-2xy^{2}-z^{2}), \partial v/\partial y=3x^{2}-4xy, logo
\frac{\partial}{\partial y}\cos(v) =-\sin\bigl(v\bigr)\,(3x^{2}-4xy).
Somando,
\frac{\partial f}{\partial y} =\frac{4y^{3}}{x^{4}+y^{4}+z^{4}} -\sin\bigl(3x^{2}y-2xy^{2}-z^{2}\bigr)\,(3x^{2}-4xy).

Passo 4

Cálculo de \displaystyle \frac{\partial f}{\partial z}:
  • Em \ln(x^{4}+y^{4}+z^{4}), \partial/\partial z \tfrac{4z^{3}}{x^{4}+y^{4}+z^{4}}.
  • Em \cos(3x^{2}y-2xy^{2}-z^{2}), \partial v/\partial z=-2z, então
\frac{\partial}{\partial z}\cos(v) =-\sin\bigl(v\bigr)\,(-2z) =2z\,\sin\bigl(v\bigr).
Logo
\frac{\partial f}{\partial z} =\frac{4z^{3}}{x^{4}+y^{4}+z^{4}} +2z\,\sin\bigl(3x^{2}y-2xy^{2}-z^{2}\bigr).

Resposta

\frac{\partial f}{\partial x} =\frac{4x^{3}}{x^{4}+y^{4}+z^{4}} -\sin\bigl(3x^{2}y-2xy^{2}-z^{2}\bigr)\,(6xy-2y^{2}),
\frac{\partial f}{\partial y} =\frac{4y^{3}}{x^{4}+y^{4}+z^{4}} -\sin\bigl(3x^{2}y-2xy^{2}-z^{2}\bigr)\,(3x^{2}-4xy),
\frac{\partial f}{\partial z} =\frac{4z^{3}}{x^{4}+y^{4}+z^{4}} +2z\,\sin\bigl(3x^{2}y-2xy^{2}-z^{2}\bigr).

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