derivative (sqrt(x))/(3(x^2+1))

解

導関数

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

\frac{- 3 x ^{2} + 1}{6 x ( x ^{2} + 1 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x}{3 ( x ^{2} + 1 )})

定数を除去:

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

= \frac{1}{3} \frac{d}{d x} (\frac{x}{x ^{2} + 1})

商の法則を適用:

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

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

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

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

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

簡素化

\frac{1}{3} \cdot \frac{\frac{1}{2 x} ( x ^{2} + 1 ) - 2 x x}{( x ^{2} + 1 ) ^{2}} : \frac{- 3 x ^{2} + 1}{6 x ( x ^{2} + 1 ) ^{2}}

= \frac{- 3 x ^{2} + 1}{6 x ( x ^{2} + 1 ) ^{2}}