Question

a derivada parcial sucessiva de segunda ordem, em relação a x, da função z= f(x,y)= sen (x³ y³) é:

Answer 1

$\displaystyle \frac{\partial^2}{\partial x^2}f(x,y)=\frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}f(x,y)\right)\\\\ \frac{\partial}{\partial x}f(x,y)=\frac{\partial}{\partial x}\sin (x^3y^3)\longrightarrow x^3y^3=u\longrightarrow \frac{\partial}{\partial x}\sin(x^3y^3)=\\\\ \frac{\partial}{\partial u}\sin(u)\cdot\frac{\partial}{\partial x}x^3y^3=\cos(u)\cdot 3x^2y^3=\boxed{3x^2y^3\cos(x^3y^3)}$
$\displaystyle \frac{\partial}{\partial x}\left(3x^2y^3\cdot\cos(x^3y^3)\right)=\frac{\partial}{\partial x}3x^2y^3\cdot\cos(x^3y^3)+\frac{\partial}{\partial x}\cos(x^3y^3)\cdot3x^2y^3\\\\\frac{\partial^2}{\partial x^2}(3x^2y^3\cdot \cos(x^3y^3))=6xy^3\cos(x^3y^3)-3x^2y^3\cdot\sin(x^3y^3)\cdot3x^2y^3\\\\ \frac{\partial^2}{\partial x^2}\left(\cos(x^3y^3)\right)=\boxed{6xy^3\cdot\cos(x^3y^3)-9x^4y^6\cdot\sin(x^3y^3)}$