derivative f(x)=(sin(x))/(x^3)

解

導関数

f (x) = \frac{sin ( x )}{x ^{3}}

\frac{x cos ( x ) - 3 sin ( x )}{x ^{4}}

解答ステップ

\frac{d}{d x} (\frac{sin ( x )}{x ^{3}})

商の法則を適用:

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

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

\frac{d}{d x} (sin (x)) = cos (x)

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

= \frac{cos ( x ) x ^{3} - 3 x ^{2} sin ( x )}{( x ^{3} ) ^{2}}

簡素化

\frac{cos ( x ) x ^{3} - 3 x ^{2} sin ( x )}{( x ^{3} ) ^{2}} : \frac{x cos ( x ) - 3 sin ( x )}{x ^{4}}

= \frac{x cos ( x ) - 3 sin ( x )}{x ^{4}}