derivative (sin(t))/(4+4cos(t))

解

導関数

\frac{sin ( t )}{4 + 4 cos ( t )}

\frac{cos ( t ) ( 4 + 4 cos ( t ) ) + 4 sin ^{2} ( t )}{( 4 + 4 cos ( t ) ) ^{2}}

解答ステップ

\frac{d}{d t} (\frac{sin ( t )}{4 + 4 cos ( t )})

商の法則を適用:

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

= \frac{\frac{d}{d t} ( sin ( t ) ) ( 4 + 4 cos ( t ) ) - \frac{d}{d t} ( 4 + 4 cos ( t ) ) sin ( t )}{( 4 + 4 cos ( t ) ) ^{2}}

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

\frac{d}{d t} (4 + 4 cos (t)) = - 4 sin (t)

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

簡素化

\frac{cos ( t ) ( 4 + 4 cos ( t ) ) - ( - 4 sin ( t ) ) sin ( t )}{( 4 + 4 cos ( t ) ) ^{2}} : \frac{cos ( t ) ( 4 + 4 cos ( t ) ) + 4 sin ^{2} ( t )}{( 4 + 4 cos ( t ) ) ^{2}}

= \frac{cos ( t ) ( 4 + 4 cos ( t ) ) + 4 sin ^{2} ( t )}{( 4 + 4 cos ( t ) ) ^{2}}