1. f(x, y)=9-x^2-7 y^3 ; fx(3,1) ; fy(3,1)
  2. f(x, y)=x^2 y e^x y ; partial f / partial x(1,1) ; partial f / partial y(1,1)
  3. z=raizx^2+4 y^2 ; partial z / partial x(1,2) ; partial z / partial y(1,2)
  4. w=x^2 cos x y ; partial w / partial x(12, pi) ; partial w / partial y(12, pi)

Passo 1

O enunciado pede para calcular as derivadas parciais de quatro funções em pontos dados. Para isso vamos usar as regras básicas de diferenciação parcial: ao calcular partial/partial x tratamos y como constante e ao calcular partial/partial y tratamos x como constante, aplicando produto, cadeia e potência conforme necessário.

Passo 2

Para f(x,y)=9 - x^2 - 7y^3 temos
partial fpartial x = -2x, partial fpartial y = -21y^2
Substituindo (x,y)=(3,1) obtemos
partial fpartial x(3,1) = -2cdot 3 = -6, partial fpartial y(3,1) = -21cdot 1^2 = -21

Passo 3

Para f(x,y)=x^2 y e^xy use produto e cadeia. Seja g=x^2y e h=e^xy. Então
partial fpartial x = partial gpartial xh + gpartial hpartial x = (2xy)e^xy + x^2ycdot e^xycdot y = e^xy(2xy + x^2y^2),
partial fpartial y = partial gpartial yh + gpartial hpartial y = (x^2)e^xy + x^2ycdot e^xycdot x = e^xy(x^2 + x^3y)
Em (1,1) isso dá
partial fpartial x(1,1) = e^1(2cdot1cdot1 + 1^2cdot1^2) = 3e, partial fpartial y(1,1) = e^1(1^2 + 1^3cdot1) = 2e

Passo 4

Para z=raizx^2+4y^2=(x^2+4y^2)^1/2 usamos a regra da cadeia:
partial zpartial x = 12(x^2+4y^2)^-1/2cdot 2x = xraizx^2+4y^2,
partial zpartial y = 12(x^2+4y^2)^-1/2cdot 8y = 4yraizx^2+4y^2
Em (1,2) temos raiz1+16=raiz17, logo
partial zpartial x(1,2) = 1raiz17, partial zpartial y(1,2) = 8raiz17

Passo 5

Para w = x^2cos(xy) usamos produto e cadeia:
partial wpartial x = 2xcos(xy) + x^2l(-sin(xy)r)cdot y = 2xcos(xy) - x^2ysin(xy),
partial wpartial y = x^2l(-sin(xy)r)cdot x = -x^3sin(xy)
Em l(tfrac12,pir) temos xy=tfracpi2, cos(tfracpi2)=0, sin(tfracpi2)=1, então
partial wpartial xl(tfrac12,pir) = 0 - l(tfrac12r)^2picdot1 = -pi4, partial wpartial yl(tfrac12,pir) = -l(tfrac12r)^3cdot1 = -18

Resposta

  1. partial fpartial x(3,1) = -6, partial fpartial y(3,1) = -21
  2. partial fpartial x(1,1) = 3e, partial fpartial y(1,1) = 2e
  3. partial zpartial x(1,2) = 1raiz17, partial zpartial y(1,2) = 8raiz17
  4. partial wpartial xl(tfrac12,pir) = -pi4, partial wpartial yl(tfrac12,pir) = -18

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