derivative 3/5 x^{5/3}-3/4 x^{1/3}+9

解

導関数

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

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

解答ステップ

\frac{d}{d x} (\frac{3}{5} x^{\frac{5}{3}} - \frac{3}{4} x^{\frac{1}{3}} + 9)

簡素化

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

= \frac{d}{d x} (\frac{3 x ^{\frac{5}{3}}}{5} - \frac{3 x ^{\frac{1}{3}}}{4} + 9)

和/差の法則を適用:

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

= \frac{d}{d x} (\frac{3 x ^{\frac{5}{3}}}{5}) - \frac{d}{d x} (\frac{3 x ^{\frac{1}{3}}}{4}) + \frac{d}{d x} (9)

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

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

\frac{d}{d x} (9) = 0

= x^{\frac{2}{3}} - \frac{1}{4 x ^{\frac{2}{3}}} + 0

簡素化

x^{\frac{2}{3}} - \frac{1}{4 x ^{\frac{2}{3}}} + 0 : \frac{- 1 + 4 x ^{\frac{4}{3}}}{4 x ^{\frac{2}{3}}}

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