derivative h(x)=(5-5x^3)^3+1/(5-5x^3)

解

導関数

h (x) = (5 - 5 x^{3})^{3} + \frac{1}{5 - 5 x ^{3}}

\frac{- 5625 x ^{14} + 22500 x ^{11} - 33750 x ^{8} + 22500 x ^{5} - 5622 x ^{2}}{5 ( x ^{3} - 1 ) ^{2}}

解答ステップ

\frac{d}{d x} ((5 - 5 x^{3})^{3} + \frac{1}{5 - 5 x ^{3}})

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} ((5 - 5 x^{3})^{3}) + \frac{d}{d x} (\frac{1}{5 - 5 x ^{3}})

\frac{d}{d x} ((5 - 5 x^{3})^{3}) = - 45 x^{2} (5 - 5 x^{3})^{2}

\frac{d}{d x} (\frac{1}{5 - 5 x ^{3}}) = \frac{3 x ^{2}}{5 ( x ^{3} - 1 ) ^{2}}

= - 45 x^{2} (5 - 5 x^{3})^{2} + \frac{3 x ^{2}}{5 ( x ^{3} - 1 ) ^{2}}

簡素化

- 45 x^{2} (5 - 5 x^{3})^{2} + \frac{3 x ^{2}}{5 ( x ^{3} - 1 ) ^{2}} : \frac{- 5625 x ^{14} + 22500 x ^{11} - 33750 x ^{8} + 22500 x ^{5} - 5622 x ^{2}}{5 ( x ^{3} - 1 ) ^{2}}

= \frac{- 5625 x ^{14} + 22500 x ^{11} - 33750 x ^{8} + 22500 x ^{5} - 5622 x ^{2}}{5 ( x ^{3} - 1 ) ^{2}}