derivative ((t^3)/(t^8+7))^2

解

導関数

(\frac{t ^{3}}{t ^{8} + 7})^{2}

\frac{2 t ^{3} ( - 5 t ^{10} + 21 t ^{2} )}{( t ^{8} + 7 ) ^{3}}

解答ステップ

\frac{d}{d t} ((\frac{t ^{3}}{t ^{8} + 7})^{2})

連鎖法則を適用:

2 \cdot \frac{t ^{3}}{t ^{8} + 7} \frac{d}{d t} (\frac{t ^{3}}{t ^{8} + 7})

= 2 \cdot \frac{t ^{3}}{t ^{8} + 7} \frac{d}{d t} (\frac{t ^{3}}{t ^{8} + 7})

\frac{d}{d t} (\frac{t ^{3}}{t ^{8} + 7}) = \frac{- 5 t ^{10} + 21 t ^{2}}{( t ^{8} + 7 ) ^{2}}

= 2 \cdot \frac{t ^{3}}{t ^{8} + 7} \cdot \frac{- 5 t ^{10} + 21 t ^{2}}{( t ^{8} + 7 ) ^{2}}

簡素化

2 \cdot \frac{t ^{3}}{t ^{8} + 7} \cdot \frac{- 5 t ^{10} + 21 t ^{2}}{( t ^{8} + 7 ) ^{2}} : \frac{2 t ^{3} ( - 5 t ^{10} + 21 t ^{2} )}{( t ^{8} + 7 ) ^{3}}

= \frac{2 t ^{3} ( - 5 t ^{10} + 21 t ^{2} )}{( t ^{8} + 7 ) ^{3}}