(\partial)/(\partial x)(({f}(x,y))(xy-y^2))

解

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

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

解答ステップ

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

y

を定数として扱う

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = f (x, y), g = x y - y^{2}

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

\frac{\partial}{\partial x} (x y - y^{2}) = y

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