derivative f(x)=(7x)/(3+x^2)

解

導関数

f (x) = \frac{7 x}{3 + x ^{2}}

\frac{7 ( 3 - x ^{2} )}{( 3 + x ^{2} ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{7 x}{3 + x ^{2}})

定数を除去:

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

= 7 \frac{d}{d x} (\frac{x}{3 + x ^{2}})

商の法則を適用:

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

= 7 \cdot \frac{\frac{d x}{d x} ( 3 + x ^{2} ) - \frac{d}{d x} ( 3 + x ^{2} ) x}{( 3 + x ^{2} ) ^{2}}

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

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

= 7 \cdot \frac{1 \cdot ( 3 + x ^{2} ) - 2 xx}{( 3 + x ^{2} ) ^{2}}

簡素化

7 \cdot \frac{1 \cdot ( 3 + x ^{2} ) - 2 xx}{( 3 + x ^{2} ) ^{2}} : \frac{7 ( 3 - x ^{2} )}{( 3 + x ^{2} ) ^{2}}

= \frac{7 ( 3 - x ^{2} )}{( 3 + x ^{2} ) ^{2}}