(\partial)/(\partial u)(v/(1+uvw))

解

\frac{\partial}{\partial u} (\frac{v}{1 + uv w})

- \frac{v ^{2} w}{( 1 + vu w ) ^{2}}

解答ステップ

\frac{\partial}{\partial u} (\frac{v}{1 + uv w})

v, w

を定数として扱う

定数を除去:

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

= v \frac{\partial}{\partial u} (\frac{1}{1 + uv w})

指数の規則を適用する:

\frac{1}{a} = a^{- 1}

= v \frac{\partial}{\partial u} ((1 + uv w)^{- 1})

連鎖法則を適用:

- \frac{1}{( 1 + uv w ) ^{2}} \frac{\partial}{\partial u} (1 + uv w)

= - \frac{1}{( 1 + uv w ) ^{2}} \frac{\partial}{\partial u} (1 + uv w)

\frac{\partial}{\partial u} (1 + uv w) = v w

= v (- \frac{1}{( 1 + uv w ) ^{2}} v w)

簡素化

v (- \frac{1}{( 1 + uv w ) ^{2}} v w) : - \frac{v ^{2} w}{( 1 + vu w ) ^{2}}

= - \frac{v ^{2} w}{( 1 + vu w ) ^{2}}