derivative (sin(2x))/(e^{x^2)}

解

導関数

\frac{sin ( 2 x )}{e ^{x^{2}}}

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

解答ステップ

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

\frac{d}{d x} (sin (2 x)) = cos (2 x) \cdot 2

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

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

簡素化

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

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