derivative of (cos(x)/(x^8))

解

\frac{d}{d x} (\frac{cos ( x )}{x ^{8}})

- \frac{x sin ( x ) + 8 cos ( x )}{x ^{9}}

解答ステップ

\frac{d}{d x} (\frac{cos ( x )}{x ^{8}})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( cos ( x ) ) x ^{8} - \frac{d}{d x} ( x ^{8} ) cos ( x )}{( x ^{8} ) ^{2}}

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

\frac{d}{d x} (x^{8}) = 8 x^{7}

= \frac{( - sin ( x ) ) x ^{8} - 8 x ^{7} cos ( x )}{( x ^{8} ) ^{2}}

簡素化

\frac{( - sin ( x ) ) x ^{8} - 8 x ^{7} cos ( x )}{( x ^{8} ) ^{2}} : - \frac{x sin ( x ) + 8 cos ( x )}{x ^{9}}

= - \frac{x sin ( x ) + 8 cos ( x )}{x ^{9}}