(\partial)/(\partial x)(f(y,x)(y/x))

解

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

y (\frac{\partial}{\partial x} (f (y, x)) \frac{1}{x} + (- \frac{1}{x ^{2}}) f (y, x))

解答ステップ

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

y

を定数として扱う

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= y \frac{\partial}{\partial x} (f (y, x) \frac{1}{x})

積の法則を適用:

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

f = f (y, x), g = \frac{1}{x}

= y (\frac{\partial}{\partial x} (f (y, x)) \frac{1}{x} + \frac{\partial}{\partial x} (\frac{1}{x}) f (y, x))

\frac{\partial}{\partial x} (\frac{1}{x}) = - \frac{1}{x ^{2}}

= y (\frac{\partial}{\partial x} (f (y, x)) \frac{1}{x} + (- \frac{1}{x ^{2}}) f (y, x))