(\partial)/(\partial x)(1/(2sqrt(1+x+y)))

解

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

- \frac{1}{4 ( 1 + x + y ) ^{\frac{3}{2}}}

解答ステップ

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

y

を定数として扱う

定数を除去:

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

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

指数の規則を適用する:

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

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

連鎖法則を適用:

- \frac{1}{2 ( 1 + x + y ) ^{\frac{3}{2}}} \frac{\partial}{\partial x} (1 + x + y)

= - \frac{1}{2 ( 1 + x + y ) ^{\frac{3}{2}}} \frac{\partial}{\partial x} (1 + x + y)

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

= \frac{1}{2} (- \frac{1}{2 ( 1 + x + y ) ^{\frac{3}{2}}} \cdot 1)

簡素化

\frac{1}{2} (- \frac{1}{2 ( 1 + x + y ) ^{\frac{3}{2}}} \cdot 1) : - \frac{1}{4 ( 1 + x + y ) ^{\frac{3}{2}}}

= - \frac{1}{4 ( 1 + x + y ) ^{\frac{3}{2}}}