Question

quais são as derivadas parciais de segunda ordem da função: g(x)= x²y+cosy+ysenx

Answer 1

$\displaystyle \frac{\partial}{\partial x}x^2y+\cos y+y\sin x=2xy+y\cos x\\\\\frac{\partial^2}{\partial x^2}=\frac{\partial}{\partial x}2xy+y\cos x=\boxed{2y-y\sin x}$ (em relação a x)
$\displaystyle \frac{\partial}{\partial y}x^2y+cos\ y+y \sin x=x^2-\sin(y)+\sin(x)\\\\\frac{\partial^2}{\partial y^2}=\frac{\partial}{\partial y}x^2-\sin(y)+\sin(x)=\boxed{-\cos(y)}$
Ou seja:
$\boxed{\displaystyle \frac{\partial^2g}{\partial x^2}=2y-\sin(x)}\\\\\boxed{\frac{\partial^2g}{\partial y^2}=-\cos(y)}$ são as derivadas parciais de segunda ordem de g(x,y)