derivative of x/(sqrt(1+5x^2))

解

\frac{d}{d x} (\frac{x}{1 + 5 x ^{2}})

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

解答ステップ

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

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

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

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

簡素化

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

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