derivative f(x)= x/(x^2+9)

解

導関数

f (x) = \frac{x}{x ^{2} + 9}

\frac{- x ^{2} + 9}{( x ^{2} + 9 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x}{x ^{2} + 9})

商の法則を適用:

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

= \frac{\frac{d x}{d x} ( x ^{2} + 9 ) - \frac{d}{d x} ( x ^{2} + 9 ) x}{( x ^{2} + 9 ) ^{2}}

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

\frac{d}{d x} (x^{2} + 9) = 2 x

= \frac{1 \cdot ( x ^{2} + 9 ) - 2 xx}{( x ^{2} + 9 ) ^{2}}

1 \cdot (x^{2} + 9) - 2 xx = - x^{2} + 9

= \frac{- x ^{2} + 9}{( x ^{2} + 9 ) ^{2}}