derivative y=(8t)/(8+sqrt(t))

解

導関数

y = \frac{8 t}{8 + t}

\frac{4 ( t + 16 )}{( 8 + t ) ^{2}}

解答ステップ

\frac{d}{d t} (\frac{8 t}{8 + t})

定数を除去:

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

= 8 \frac{d}{d t} (\frac{t}{8 + t})

商の法則を適用:

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

= 8 \cdot \frac{\frac{d t}{d t} ( 8 + t ) - \frac{d}{d t} ( 8 + t ) t}{( 8 + t ) ^{2}}

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

\frac{d}{d t} (8 + t) = \frac{1}{2 t}

= 8 \cdot \frac{1 \cdot ( 8 + t ) - \frac{1}{2 t} t}{( 8 + t ) ^{2}}

簡素化

8 \cdot \frac{1 \cdot ( 8 + t ) - \frac{1}{2 t} t}{( 8 + t ) ^{2}} : \frac{4 ( t + 16 )}{( 8 + t ) ^{2}}

= \frac{4 ( t + 16 )}{( 8 + t ) ^{2}}