(\partial)/(\partial x)(xy-3f(x,y)-f(x,y)x^2+f(x,y)y)

解

\frac{\partial}{\partial x} (x y - 3 f (x, y) - f (x, y) x^{2} + f (x, y) y)

y - 3 \frac{\partial}{\partial x} (f (x, y)) - (\frac{\partial}{\partial x} (f (x, y)) x^{2} + 2 x f (x, y)) + y \frac{\partial}{\partial x} (f (x, y))

解答ステップ

\frac{\partial}{\partial x} (x y - 3 f (x, y) - f (x, y) x^{2} + f (x, y) y)

y

を定数として扱う

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{\partial}{\partial x} (x y) - \frac{\partial}{\partial x} (3 f (x, y)) - \frac{\partial}{\partial x} (f (x, y) x^{2}) + \frac{\partial}{\partial x} (f (x, y) y)

\frac{\partial}{\partial x} (x y) = y

\frac{\partial}{\partial x} (3 f (x, y)) = 3 \frac{\partial}{\partial x} (f (x, y))

\frac{\partial}{\partial x} (f (x, y) x^{2}) = \frac{\partial}{\partial x} (f (x, y)) x^{2} + 2 x f (x, y)

\frac{\partial}{\partial x} (f (x, y) y) = y \frac{\partial}{\partial x} (f (x, y))

= y - 3 \frac{\partial}{\partial x} (f (x, y)) - (\frac{\partial}{\partial x} (f (x, y)) x^{2} + 2 x f (x, y)) + y \frac{\partial}{\partial x} (f (x, y))