derivative of (cos(12x)/(e^{7x)})

解

\frac{d}{d x} (\frac{cos ( 12 x )}{e ^{7 x}})

- \frac{12 sin ( 12 x ) + 7 cos ( 12 x )}{e ^{7 x}}

解答ステップ

\frac{d}{d x} (\frac{cos ( 12 x )}{e ^{7 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} ( cos ( 12 x ) ) e ^{7 x} - \frac{d}{d x} ( e ^{7 x} ) cos ( 12 x )}{( e ^{7 x} ) ^{2}}

\frac{d}{d x} (cos (12 x)) = - sin (12 x) \cdot 12

\frac{d}{d x} (e^{7 x}) = e^{7 x} \cdot 7

= \frac{( - sin ( 12 x ) \cdot 12 ) e ^{7 x} - e ^{7 x} \cdot 7 cos ( 12 x )}{( e ^{7 x} ) ^{2}}

簡素化

\frac{( - sin ( 12 x ) \cdot 12 ) e ^{7 x} - e ^{7 x} \cdot 7 cos ( 12 x )}{( e ^{7 x} ) ^{2}} : - \frac{12 sin ( 12 x ) + 7 cos ( 12 x )}{e ^{7 x}}

= - \frac{12 sin ( 12 x ) + 7 cos ( 12 x )}{e ^{7 x}}