derivative of (-4x^6+4x^5-3x^3/(3x^2))

解

\frac{d}{d x} (\frac{- 4 x ^{6} + 4 x ^{5} - 3 x ^{3}}{3 x ^{2}})

- \frac{16 x ^{3}}{3} + 4 x^{2} - 1

解答ステップ

\frac{d}{d x} (\frac{- 4 x ^{6} + 4 x ^{5} - 3 x ^{3}}{3 x ^{2}})

定数を除去:

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

= \frac{1}{3} \frac{d}{d x} (\frac{- 4 x ^{6} + 4 x ^{5} - 3 x ^{3}}{x ^{2}})

商の法則を適用:

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

= \frac{1}{3} \cdot \frac{\frac{d}{d x} ( - 4 x ^{6} + 4 x ^{5} - 3 x ^{3} ) x ^{2} - \frac{d}{d x} ( x ^{2} ) ( - 4 x ^{6} + 4 x ^{5} - 3 x ^{3} )}{( x ^{2} ) ^{2}}

\frac{d}{d x} (- 4 x^{6} + 4 x^{5} - 3 x^{3}) = - 24 x^{5} + 20 x^{4} - 9 x^{2}

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

= \frac{1}{3} \cdot \frac{( - 24 x ^{5} + 20 x ^{4} - 9 x ^{2} ) x ^{2} - 2 x ( - 4 x ^{6} + 4 x ^{5} - 3 x ^{3} )}{( x ^{2} ) ^{2}}

簡素化

\frac{1}{3} \cdot \frac{( - 24 x ^{5} + 20 x ^{4} - 9 x ^{2} ) x ^{2} - 2 x ( - 4 x ^{6} + 4 x ^{5} - 3 x ^{3} )}{( x ^{2} ) ^{2}} : - \frac{16 x ^{3}}{3} + 4 x^{2} - 1

= - \frac{16 x ^{3}}{3} + 4 x^{2} - 1