derivative of (sin(x-x)/(x^3))

解

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

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

解答ステップ

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

\frac{d}{d x} (sin (x) - x) = cos (x) - 1

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

= \frac{( cos ( x ) - 1 ) x ^{3} - 3 x ^{2} ( sin ( x ) - x )}{( x ^{3} ) ^{2}}

簡素化

\frac{( cos ( x ) - 1 ) x ^{3} - 3 x ^{2} ( sin ( x ) - x )}{( x ^{3} ) ^{2}} : \frac{2 x + x cos ( x ) - 3 sin ( x )}{x ^{4}}

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