(\partial)/(\partial x)(((xv^2))/((x^2+v^2)))

解

\frac{\partial}{\partial x} (\frac{( x v ^{2} )}{( x ^{2} + v ^{2} )})

\frac{v ^{2} ( - x ^{2} + v ^{2} )}{( x ^{2} + v ^{2} ) ^{2}}

解答ステップ

\frac{\partial}{\partial x} (\frac{( x v ^{2} )}{( x ^{2} + v ^{2} )})

v

を定数として扱う

定数を除去:

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

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

商の法則を適用:

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

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

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

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

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

簡素化

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

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