derivative ((t^3)/(t^6+9))^2

解

導関数

(\frac{t ^{3}}{t ^{6} + 9})^{2}

\frac{2 t ^{3} ( - 3 t ^{8} + 27 t ^{2} )}{( t ^{6} + 9 ) ^{3}}

解答ステップ

\frac{d}{d t} ((\frac{t ^{3}}{t ^{6} + 9})^{2})

連鎖法則を適用:

2 \cdot \frac{t ^{3}}{t ^{6} + 9} \frac{d}{d t} (\frac{t ^{3}}{t ^{6} + 9})

= 2 \cdot \frac{t ^{3}}{t ^{6} + 9} \frac{d}{d t} (\frac{t ^{3}}{t ^{6} + 9})

\frac{d}{d t} (\frac{t ^{3}}{t ^{6} + 9}) = \frac{- 3 t ^{8} + 27 t ^{2}}{( t ^{6} + 9 ) ^{2}}

= 2 \cdot \frac{t ^{3}}{t ^{6} + 9} \cdot \frac{- 3 t ^{8} + 27 t ^{2}}{( t ^{6} + 9 ) ^{2}}

簡素化

2 \cdot \frac{t ^{3}}{t ^{6} + 9} \cdot \frac{- 3 t ^{8} + 27 t ^{2}}{( t ^{6} + 9 ) ^{2}} : \frac{2 t ^{3} ( - 3 t ^{8} + 27 t ^{2} )}{( t ^{6} + 9 ) ^{3}}

= \frac{2 t ^{3} ( - 3 t ^{8} + 27 t ^{2} )}{( t ^{6} + 9 ) ^{3}}