derivative of x/(sqrt(5+x^2))

解

\frac{d}{d x} (\frac{x}{5 + x ^{2}})

\frac{5}{( 5 + x ^{2} ) 5 + x ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x}{5 + x ^{2}})

商の法則を適用:

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

= \frac{\frac{d x}{d x} 5 + x ^{2} - \frac{d}{d x} ( 5 + x ^{2} ) x}{( 5 + x ^{2} ) ^{2}}

\frac{d x}{d x} = 1

\frac{d}{d x} (5 + x^{2}) = \frac{x}{5 + x ^{2}}

= \frac{1 \cdot 5 + x ^{2} - \frac{x}{5 + x ^{2}} x}{( 5 + x ^{2} ) ^{2}}

簡素化

\frac{1 \cdot 5 + x ^{2} - \frac{x}{5 + x ^{2}} x}{( 5 + x ^{2} ) ^{2}} : \frac{5}{( 5 + x ^{2} ) 5 + x ^{2}}

= \frac{5}{( 5 + x ^{2} ) 5 + x ^{2}}