derivative (sqrt(t))/(1+t^2)

解

導関数

\frac{t}{1 + t ^{2}}

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

解答ステップ

\frac{d}{d t} (\frac{t}{1 + t ^{2}})

商の法則を適用:

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

= \frac{\frac{d}{d t} ( t ) ( 1 + t ^{2} ) - \frac{d}{d t} ( 1 + t ^{2} ) t}{( 1 + t ^{2} ) ^{2}}

\frac{d}{d t} (t) = \frac{1}{2 t}

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

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

簡素化

\frac{\frac{1}{2 t} ( 1 + t ^{2} ) - 2 t t}{( 1 + t ^{2} ) ^{2}} : \frac{1 - 3 t ^{2}}{2 t ( 1 + t ^{2} ) ^{2}}

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