derivative of ({g}(x(sqrt(x)))/(ax^2+x))

解

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

\frac{2 a x ^{2} x \frac{d}{d x} ( g ( x ) ) - 4 a x x g ( x ) + 2 x x \frac{d}{d x} ( g ( x ) ) + a x ^{\frac{3}{2}} g ( x ) - x g ( x )}{2 ( a x ^{2} + x ) ^{2}}

解答ステップ

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

商の法則を適用:

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

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

\frac{d}{d x} (g (x) x) = x \frac{d}{d x} (g (x)) + \frac{g ( x )}{2 x}

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

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

簡素化

\frac{( x \frac{d}{d x} ( g ( x ) ) + \frac{g ( x )}{2 x} ) ( a x ^{2} + x ) - ( 2 a x + 1 ) g ( x ) x}{( a x ^{2} + x ) ^{2}} : \frac{2 a x ^{2} x \frac{d}{d x} ( g ( x ) ) - 4 a x x g ( x ) + 2 x x \frac{d}{d x} ( g ( x ) ) + a x ^{\frac{3}{2}} g ( x ) - x g ( x )}{2 ( a x ^{2} + x ) ^{2}}

= \frac{2 a x ^{2} x \frac{d}{d x} ( g ( x ) ) - 4 a x x g ( x ) + 2 x x \frac{d}{d x} ( g ( x ) ) + a x ^{\frac{3}{2}} g ( x ) - x g ( x )}{2 ( a x ^{2} + x ) ^{2}}