derivative (cos(t))/(t^7)

解

導関数

\frac{cos ( t )}{t ^{7}}

- \frac{t sin ( t ) + 7 cos ( t )}{t ^{8}}

解答ステップ

\frac{d}{d t} (\frac{cos ( t )}{t ^{7}})

商の法則を適用:

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

= \frac{\frac{d}{d t} ( cos ( t ) ) t ^{7} - \frac{d}{d t} ( t ^{7} ) cos ( t )}{( t ^{7} ) ^{2}}

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

\frac{d}{d t} (t^{7}) = 7 t^{6}

= \frac{( - sin ( t ) ) t ^{7} - 7 t ^{6} cos ( t )}{( t ^{7} ) ^{2}}

簡素化

\frac{( - sin ( t ) ) t ^{7} - 7 t ^{6} cos ( t )}{( t ^{7} ) ^{2}} : - \frac{t sin ( t ) + 7 cos ( t )}{t ^{8}}

= - \frac{t sin ( t ) + 7 cos ( t )}{t ^{8}}