derivative of (-3x^4-5x^3+2/(-5x^4))

解

\frac{d}{d x} (\frac{- 3 x ^{4} - 5 x ^{3} + 2}{- 5 x ^{4}})

- \frac{5 x ^{3} - 8}{5 x ^{5}}

解答ステップ

\frac{d}{d x} (\frac{- 3 x ^{4} - 5 x ^{3} + 2}{- 5 x ^{4}})

簡素化

\frac{- 3 x ^{4} - 5 x ^{3} + 2}{- 5 x ^{4}} : - \frac{- 3 x ^{4} - 5 x ^{3} + 2}{5 x ^{4}}

= \frac{d}{d x} (- \frac{- 3 x ^{4} - 5 x ^{3} + 2}{5 x ^{4}})

定数を除去:

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

= - \frac{1}{5} \frac{d}{d x} (\frac{- 3 x ^{4} - 5 x ^{3} + 2}{x ^{4}})

商の法則を適用:

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

= - \frac{1}{5} \cdot \frac{\frac{d}{d x} ( - 3 x ^{4} - 5 x ^{3} + 2 ) x ^{4} - \frac{d}{d x} ( x ^{4} ) ( - 3 x ^{4} - 5 x ^{3} + 2 )}{( x ^{4} ) ^{2}}

\frac{d}{d x} (- 3 x^{4} - 5 x^{3} + 2) = - 12 x^{3} - 15 x^{2}

\frac{d}{d x} (x^{4}) = 4 x^{3}

= - \frac{1}{5} \cdot \frac{( - 12 x ^{3} - 15 x ^{2} ) x ^{4} - 4 x ^{3} ( - 3 x ^{4} - 5 x ^{3} + 2 )}{( x ^{4} ) ^{2}}

簡素化

- \frac{1}{5} \cdot \frac{( - 12 x ^{3} - 15 x ^{2} ) x ^{4} - 4 x ^{3} ( - 3 x ^{4} - 5 x ^{3} + 2 )}{( x ^{4} ) ^{2}} : - \frac{5 x ^{3} - 8}{5 x ^{5}}

= - \frac{5 x ^{3} - 8}{5 x ^{5}}