derivative of (e^x+cos(x)/(x^2))

解

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

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

解答ステップ

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

商の法則を適用:

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

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

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

\frac{d}{d x} (x^{2}) = 2 x

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

簡素化

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

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