derivative f(t)=(2t)/(2+sqrt(t))

解

導関数

f (t) = \frac{2 t}{2 + t}

\frac{4 + t}{( 2 + t ) ^{2}}

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

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

\frac{d t}{d t} = 1

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

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

簡素化

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

= \frac{4 + t}{( 2 + t ) ^{2}}