(\partial)/(\partial x)(x/(x+y/x))

解

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

\frac{2 x y}{( x ^{2} + y ) ^{2}}

解答ステップ

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

y

を定数として扱う

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

\frac{\partial}{\partial x} (x) = 1

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

= \frac{1 \cdot ( x + \frac{y}{x} ) - ( 1 - \frac{y}{x ^{2}} ) x}{( x + \frac{y}{x} ) ^{2}}

簡素化

\frac{1 \cdot ( x + \frac{y}{x} ) - ( 1 - \frac{y}{x ^{2}} ) x}{( x + \frac{y}{x} ) ^{2}} : \frac{2 x y}{( x ^{2} + y ) ^{2}}

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