derivative 1/8 x^{2/3}(9x^2-8x-16)

解

導関数

\frac{1}{8} x^{\frac{2}{3}} (9 x^{2} - 8 x - 16)

\frac{9 x ^{2} - 5 x - 4}{3 x ^{\frac{1}{3}}}

解答ステップ

\frac{d}{d x} (\frac{1}{8} x^{\frac{2}{3}} (9 x^{2} - 8 x - 16))

定数を除去:

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

= \frac{1}{8} \frac{d}{d x} (x^{\frac{2}{3}} (9 x^{2} - 8 x - 16))

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{\frac{2}{3}}, g = 9 x^{2} - 8 x - 16

= \frac{1}{8} (\frac{d}{d x} (x^{\frac{2}{3}}) (9 x^{2} - 8 x - 16) + \frac{d}{d x} (9 x^{2} - 8 x - 16) x^{\frac{2}{3}})

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

\frac{d}{d x} (9 x^{2} - 8 x - 16) = 18 x - 8

= \frac{1}{8} (\frac{2}{3 x ^{\frac{1}{3}}} (9 x^{2} - 8 x - 16) + (18 x - 8) x^{\frac{2}{3}})

簡素化

\frac{1}{8} (\frac{2}{3 x ^{\frac{1}{3}}} (9 x^{2} - 8 x - 16) + (18 x - 8) x^{\frac{2}{3}}) : \frac{9 x ^{2} - 5 x - 4}{3 x ^{\frac{1}{3}}}

= \frac{9 x ^{2} - 5 x - 4}{3 x ^{\frac{1}{3}}}