Question

Calcule as derivadas de segunda ordem da função
f(x,y)=x3y− 2y2−2−4

Answer 1

Olá Estudante!

Contendo a relação:

$\huge {\boxed {\sf \bf f \left ( x,y \right ) = x^3y- 2y^2-2-4 }}$

✍ Primeiramente iremos derivar x e posteriormente y, adotando as seguintes propriedades:

$\huge {\boxed {\boxed {\green {\sf f_{xx} = \left ( f_x \right ) _x = \cfrac{ \partial }{ \partial x} \left ( \cfrac{\partial f}{\partial x} \right ) = \cfrac{\partial ^2 f}{\partial x^2} }}}}$

$\huge {\boxed {\boxed {\blue {\sf f_{xy} = \left ( f_x \right )_y = \cfrac{\partial }{\partial y} \left ( \cfrac{\partial f}{\partial x} \right ) = \cfrac{\partial ^2f}{\partial y\partial x} }}}}$

$\huge {\boxed {\boxed {\red {\sf f_{yy} = \left ( f_x\right ) _y = \cfrac{\partial }{ \partial y} \left ( \cfrac{\partial f}{\partial y} \right ) = \cfrac{\partial ^2f}{\partial y^2} }}}}$

$\huge {\boxed {\boxed {\sf \bf f_{yx} = \left ( f_y \right )_x = \cfrac{\partial }{\partial x} \left ( \cfrac{\partial f}{\partial y} \right ) = \cfrac{\partial ^2f }{\partial x \partial y} }}}$

➡️ Portanto:

$\huge {\boxed {\gray {\sf f_x(x,y) = 3xy }}}$

$\huge {\boxed {\blue {\sf f_{xx} \left ( x,y \right ) = 3y }}}$

$\huge {\boxed{ \purple {\sf f_y \left ( x,y \right ) = x^3 - 4y }}}$

$\huge {\boxed {\pink {\sf f_{yy} \left ( x,y \right ) = -4 }}}$

$\huge {\boxed {\sf \bf f_{xy} \left ( x,y \right ) = 3x }}$

$\huge {\boxed {\sf \bf f_{yx} \left ( x,y \right ) = 3x }}$

• Att. MatiasHP