derivative of (sin(8x)/(e^{4x)})

解

\frac{d}{d x} (\frac{sin ( 8 x )}{e ^{4 x}})

\frac{4 ( 2 cos ( 8 x ) - sin ( 8 x ) )}{e ^{4 x}}

解答ステップ

\frac{d}{d x} (\frac{sin ( 8 x )}{e ^{4 x}})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( sin ( 8 x ) ) e ^{4 x} - \frac{d}{d x} ( e ^{4 x} ) sin ( 8 x )}{( e ^{4 x} ) ^{2}}

\frac{d}{d x} (sin (8 x)) = cos (8 x) \cdot 8

\frac{d}{d x} (e^{4 x}) = e^{4 x} \cdot 4

= \frac{cos ( 8 x ) \cdot 8 e ^{4 x} - e ^{4 x} \cdot 4 sin ( 8 x )}{( e ^{4 x} ) ^{2}}

簡素化

\frac{cos ( 8 x ) \cdot 8 e ^{4 x} - e ^{4 x} \cdot 4 sin ( 8 x )}{( e ^{4 x} ) ^{2}} : \frac{4 ( 2 cos ( 8 x ) - sin ( 8 x ) )}{e ^{4 x}}

= \frac{4 ( 2 cos ( 8 x ) - sin ( 8 x ) )}{e ^{4 x}}