derivative ((e^x(2x+3)))/(x^2+1)

解

導関数

\frac{( e ^{x} ( 2 x + 3 ) )}{x ^{2} + 1}

\frac{2 e ^{x} x ^{3} + e ^{x} x ^{2} - 4 e ^{x} x + 5 e ^{x}}{( x ^{2} + 1 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{e ^{x} ( 2 x + 3 )}{x ^{2} + 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} ( e ^{x} ( 2 x + 3 ) ) ( x ^{2} + 1 ) - \frac{d}{d x} ( x ^{2} + 1 ) e ^{x} ( 2 x + 3 )}{( x ^{2} + 1 ) ^{2}}

\frac{d}{d x} (e^{x} (2 x + 3)) = 2 e^{x} x + 5 e^{x}

\frac{d}{d x} (x^{2} + 1) = 2 x

= \frac{( 2 e ^{x} x + 5 e ^{x} ) ( x ^{2} + 1 ) - 2 x e ^{x} ( 2 x + 3 )}{( x ^{2} + 1 ) ^{2}}

拡張

(2 e^{x} x + 5 e^{x}) (x^{2} + 1) - 2 x e^{x} (2 x + 3) : 2 e^{x} x^{3} + e^{x} x^{2} - 4 e^{x} x + 5 e^{x}

= \frac{2 e ^{x} x ^{3} + e ^{x} x ^{2} - 4 e ^{x} x + 5 e ^{x}}{( x ^{2} + 1 ) ^{2}}