Question

a derivada a^2f/aydx da função f(x,y)=ln(xy)+x.y^2

Answer 1

Temos então a derivada parcial em ordem a $x$:

$\dfrac{\partial f}{\partial x} = \dfrac{\partial}{\partial x} [\ln(xy)] + \dfrac{\partial}{\partial x}(xy^2) = \dfrac{1}{xy}\dfrac{\partial}{\partial x} (xy)  + y^2 = \dfrac{1}{xy}y + y^2 = \dfrac{1}{x} + y^2.$

Portanto, a derivada pretendida é:

$\dfrac{\partial^2 f}{\partial y \partial x} = \dfrac{\partial}{\partial y}\left(\dfrac{\partial f}{\partial x}\right) = \dfrac{\partial}{\partial y}\left(\dfrac{1}{x} + y^2\right) = 2y.$

Resposta: $\boxed{\dfrac{\partial^2 f}{\partial y \partial x} = 2y}.$