derivative (90t^2)/(t^2+150)

解

導関数

\frac{90 t ^{2}}{t ^{2} + 150}

\frac{27000 t}{( t ^{2} + 150 ) ^{2}}

解答ステップ

\frac{d}{d t} (\frac{90 t ^{2}}{t ^{2} + 150})

定数を除去:

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

= 90 \frac{d}{d t} (\frac{t ^{2}}{t ^{2} + 150})

商の法則を適用:

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

= 90 \cdot \frac{\frac{d}{d t} ( t ^{2} ) ( t ^{2} + 150 ) - \frac{d}{d t} ( t ^{2} + 150 ) t ^{2}}{( t ^{2} + 150 ) ^{2}}

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

\frac{d}{d t} (t^{2} + 150) = 2 t

= 90 \cdot \frac{2 t ( t ^{2} + 150 ) - 2 t t ^{2}}{( t ^{2} + 150 ) ^{2}}

簡素化

90 \cdot \frac{2 t ( t ^{2} + 150 ) - 2 t t ^{2}}{( t ^{2} + 150 ) ^{2}} : \frac{27000 t}{( t ^{2} + 150 ) ^{2}}

= \frac{27000 t}{( t ^{2} + 150 ) ^{2}}