derivative g(x)=(arcsin(3x))/x

解

導関数

g (x) = \frac{arcsin ( 3 x )}{x}

\frac{3 x - arcsin ( 3 x ) 1 - 9 x ^{2}}{x ^{2} 1 - 9 x ^{2}}

解答ステップ

\frac{d}{d x} (\frac{arcsin ( 3 x )}{x})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( arcsin ( 3 x ) ) x - \frac{d x}{d x} arcsin ( 3 x )}{x ^{2}}

\frac{d}{d x} (arcsin (3 x)) = \frac{3}{1 - 9 x ^{2}}

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

= \frac{\frac{3}{1 - 9 x ^{2}} x - 1 \cdot arcsin ( 3 x )}{x ^{2}}

簡素化

\frac{\frac{3}{1 - 9 x ^{2}} x - 1 \cdot arcsin ( 3 x )}{x ^{2}} : \frac{3 x - arcsin ( 3 x ) 1 - 9 x ^{2}}{x ^{2} 1 - 9 x ^{2}}

= \frac{3 x - arcsin ( 3 x ) 1 - 9 x ^{2}}{x ^{2} 1 - 9 x ^{2}}