derivative of (sin(x)/(bsin(bx)))

解

\frac{d}{d x} (\frac{sin ( x )}{b sin ( b x )})

\frac{sin ( b x ) cos ( x ) - b cos ( b x ) sin ( x )}{b sin ^{2} ( b x )}

解答ステップ

\frac{d}{d x} (\frac{sin ( x )}{b sin ( b x )})

定数を除去:

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

= \frac{1}{b} \frac{d}{d x} (\frac{sin ( x )}{sin ( b x )})

商の法則を適用:

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

= \frac{1}{b} \cdot \frac{\frac{d}{d x} ( sin ( x ) ) sin ( b x ) - \frac{d}{d x} ( sin ( b x ) ) sin ( x )}{( sin ( b x ) ) ^{2}}

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

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

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

簡素化

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

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