derivative f(x)=(cos(e^x))/(x^4+1)

解

導関数

f (x) = \frac{cos ( e ^{x} )}{x ^{4} + 1}

\frac{- e ^{x} sin ( e ^{x} ) ( x ^{4} + 1 ) - 4 x ^{3} cos ( e ^{x} )}{( x ^{4} + 1 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{cos ( e ^{x} )}{x ^{4} + 1})

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 ( e ^{x} ) ) ( x ^{4} + 1 ) - \frac{d}{d x} ( x ^{4} + 1 ) cos ( e ^{x} )}{( x ^{4} + 1 ) ^{2}}

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

\frac{d}{d x} (x^{4} + 1) = 4 x^{3}

= \frac{( - sin ( e ^{x} ) e ^{x} ) ( x ^{4} + 1 ) - 4 x ^{3} cos ( e ^{x} )}{( x ^{4} + 1 ) ^{2}}

簡素化

= \frac{- e ^{x} sin ( e ^{x} ) ( x ^{4} + 1 ) - 4 x ^{3} cos ( e ^{x} )}{( x ^{4} + 1 ) ^{2}}