derivative of x/(x^2+t^2)

解

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

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

解答ステップ

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

t

を定数として扱う

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

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

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

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

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

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