derivative of (x{g}(x(x)^{x-1})/(x^2+1))

解

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

\frac{( x ^{x} \frac{d}{d x} ( g ( x ) ) + x ^{x} ln ( x ) g ( x ) + x ^{x} g ( x ) ) ( x ^{2} + 1 ) - 2 x ^{x + 1} g ( x )}{( x ^{2} + 1 ) ^{2}}

解答ステップ

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

\frac{d}{d x} (x g (x) x^{x - 1}) = x^{x} \frac{d}{d x} (g (x)) + x^{x} ln (x) g (x) + x^{x} g (x)

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

= \frac{( x ^{x} \frac{d}{d x} ( g ( x ) ) + x ^{x} ln ( x ) g ( x ) + x ^{x} g ( x ) ) ( x ^{2} + 1 ) - 2 xx g ( x ) x ^{x - 1}}{( x ^{2} + 1 ) ^{2}}

2 xx g (x) x^{x - 1} = 2 x^{x + 1} g (x)

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