Question

Дана функция $f(x;y)=sin(x^2y)$ . Найти все её частные производные второго порядка.

Answer 1

Ответ:

$\frac{\partial ^2 f}{\partial x^2}=\frac{\partial}{\partial x}\left(2xy\cos{(x^2y)}\right)=2y\cos{(x^2y)}-2xy\sin{(x^2y)}\cdot2xy=2y\left(\cos{(x^2y)}-2x^2y\sin{(x^2y)}\right)\\\\\\ \frac{\partial ^2 f}{\partial x \partial y}=\frac{\partial}{\partial x}\left(x^2\cos{(x^2y)}\right)=-x^2\sin{(x^2y)}\cdot2xy+2x\cos{(x^2y)}=2x\left(\cos{(x^2y)}-x^2y\sin{(x^2y)}\right)\\\\\\ \frac{\partial ^2 f}{\partial y^2}=\frac{\partial}{\partial y}\left(x^2\cos{(x^2y)}\right)=-x^2\sin{(x^2y)}\cdot x^2=-x^4\sin{(x^2y)}\\\\\\ \frac{\partial ^2 f}{\partial y \partial x}=\frac{\partial}{\partial y}\left(2xy\cos{(x^2y)}\right)=2x\cos{(x^2y)}-x^2\sin{(x^2y)}\cdot2xy=2x\left(\cos{(x^2y)}-x^2y\sin{(x^2y)}\right)$