derivative f(x)=(4x)/((x^2+1)^2)

解

導関数

f (x) = \frac{4 x}{( x ^{2} + 1 ) ^{2}}

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

解答ステップ

\frac{d}{d x} (\frac{4 x}{( x ^{2} + 1 ) ^{2}})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 4 \frac{d}{d x} (\frac{x}{( x ^{2} + 1 ) ^{2}})

商の法則を適用:

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

= 4 \cdot \frac{\frac{d x}{d x} ( x ^{2} + 1 ) ^{2} - \frac{d}{d x} ( ( x ^{2} + 1 ) ^{2} ) x}{( ( x ^{2} + 1 ) ^{2} ) ^{2}}

\frac{d x}{d x} = 1

\frac{d}{d x} ((x^{2} + 1)^{2}) = 4 x (x^{2} + 1)

= 4 \cdot \frac{1 \cdot ( x ^{2} + 1 ) ^{2} - 4 x ( x ^{2} + 1 ) x}{( ( x ^{2} + 1 ) ^{2} ) ^{2}}

簡素化

4 \cdot \frac{1 \cdot ( x ^{2} + 1 ) ^{2} - 4 x ( x ^{2} + 1 ) x}{( ( x ^{2} + 1 ) ^{2} ) ^{2}} : \frac{4 ( - 3 x ^{2} + 1 )}{( x ^{2} + 1 ) ^{3}}

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