derivative (cos(x))/(e^x)

解

導関数

\frac{cos ( x )}{e ^{x}}

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

解答ステップ

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

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

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

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

簡素化

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

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