(x/(sqrt(x^2+1)))^'

解

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

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

解答ステップ

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

x^{'} = 1

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

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

簡素化

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

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