Calcule as derivadas de segunda ordem da função f dada por:
(b) f(x, y)=arctg(y / x)+e^-x / y+tg(y / x)

Passo 1

O enunciado pede as derivadas parciais de segunda ordem de
f(x,y)=arctan!l(tfracyxr);+;e^-x/y;+;tan!l(tfracyxr)
Precisamos primeiro encontrar as derivadas parciais de primeira ordem partial f/partial x e partial f/partial y, e então diferenciá-las novamente em x e em y.

Passo 2

Calculamos
fx=partialpartial xl[arctan(tfracyx)r] +partialpartial xl[e^-x/yr] +partialpartial xl[tan(tfracyx)r]
  1. partialpartial xarctan(tfracyx) =11+(y/x)^2cdotl(-tfracyx^2r) =-yx^2+y^2
  2. partialpartial xe^-x/y =e^-x/ycdotl(-tfrac1yr) =-1y,e^-x/y
  3. partialpartial xtan(tfracyx) =sec^2(tfracyx)cdotl(-tfracyx^2r) =-yx^2,sec^2!l(tfrac yxr)
Logo
fx =-yx^2+y^2 -1y,e^-x/y -yx^2,sec^2!l(tfrac yxr)

Passo 3

Calculamos
fy=partialpartial yl[arctan(tfracyx)r] +partialpartial yl[e^-x/yr] +partialpartial yl[tan(tfracyx)r]
  1. partialpartial yarctan(tfracyx) =11+(y/x)^2cdot1x =xx^2+y^2
  2. partialpartial ye^-x/y =e^-x/ycdotxy^2 =xy^2,e^-x/y
  3. partialpartial ytan(tfracyx) =sec^2(tfracyx)cdot1x =1x,sec^2!l(tfrac yxr)
Logo
fy =xx^2+y^2 +xy^2,e^-x/y +1x,sec^2!l(tfrac yxr)

Passo 4

Agora derivamos fx em x e em y:
fxx&=partialpartial xl[-tfracyx^2+y^2r] +partialpartial xl[-tfrac1y,e^-x/yr] +partialpartial xl[-tfracyx^2sec^2(tfrac yx)r] &=2xy(x^2+y^2)^2 +1y^2e^-x/y +2yx^3,sec^2!l(tfrac yxr) +2y^2x^4,sec^2!l(tfrac yxr)tan!l(tfrac yxr),
fxy&=partialpartial yl[-tfracyx^2+y^2r] +partialpartial yl[-tfrac1y,e^-x/yr] +partialpartial yl[-tfracyx^2sec^2(tfrac yx)r] &=y^2-x^2(x^2+y^2)^2 +y - xy^3,e^-x/y -1x^2,sec^2!l(tfrac yxr) -2yx^3,sec^2!l(tfrac yxr)tan!l(tfrac yxr)

Passo 5

Por fim derivamos fy em y:
fyy&=partialpartial yl[tfracxx^2+y^2r] +partialpartial yl[tfracxy^2e^-x/yr] +partialpartial yl[tfrac1x,sec^2(tfrac yx)r] &=-2xy(x^2+y^2)^2 -2xy^3,e^-x/y +x^2y^4,e^-x/y +2x^2,sec^2!l(tfrac yxr)tan!l(tfrac yxr)

Resposta

fxx =2xy(x^2+y^2)^2 +1y^2e^-x/y +2yx^3sec^2!l(tfrac yxr) +2y^2x^4sec^2!l(tfrac yxr)tan!l(tfrac yxr),
fxy =y^2-x^2(x^2+y^2)^2 +y - xy^3e^-x/y -1x^2sec^2!l(tfrac yxr) -2yx^3sec^2!l(tfrac yxr)tan!l(tfrac yxr),
fyy =-2xy(x^2+y^2)^2 -2xy^3e^-x/y +x^2y^4e^-x/y +2x^2sec^2!l(tfrac yxr)tan!l(tfrac yxr)

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