derivative of (3sin(x)/(3+cos(x)))

解

\frac{d}{d x} (\frac{3 sin ( x )}{3 + cos ( x )})

\frac{3 ( 1 + 3 cos ( x ) )}{( 3 + cos ( x ) ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{3 sin ( x )}{3 + cos ( x )})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 3 \frac{d}{d x} (\frac{sin ( x )}{3 + cos ( x )})

商の法則を適用:

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

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

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

\frac{d}{d x} (3 + cos (x)) = - sin (x)

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

簡素化

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

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