derivative of 1/(x+sqrt(x))

解

\frac{d}{d x} (\frac{1}{x + x})

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

解答ステップ

\frac{d}{d x} (\frac{1}{x + x})

指数の規則を適用する:

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

= \frac{d}{d x} ((x + x)^{- 1})

連鎖法則を適用:

- \frac{1}{( x + x ) ^{2}} \frac{d}{d x} (x + x)

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

\frac{d}{d x} (x + x) = 1 + \frac{1}{2 x}

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

簡素化

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

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