derivative of ((4x+1/(2-5x))^{1/3})

解

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

\frac{13}{3 ( 4 x + 1 ) ^{\frac{2}{3}} ( - 5 x + 2 ) ^{\frac{4}{3}}}

解答ステップ

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

連鎖法則を適用:

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

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

\frac{d}{d x} (\frac{4 x + 1}{2 - 5 x}) = \frac{13}{( 2 - 5 x ) ^{2}}

= \frac{1}{3 ( \frac{4 x + 1}{2 - 5 x} ) ^{\frac{2}{3}}} \cdot \frac{13}{( 2 - 5 x ) ^{2}}

簡素化

\frac{1}{3 ( \frac{4 x + 1}{2 - 5 x} ) ^{\frac{2}{3}}} \cdot \frac{13}{( 2 - 5 x ) ^{2}} : \frac{13}{3 ( 4 x + 1 ) ^{\frac{2}{3}} ( - 5 x + 2 ) ^{\frac{4}{3}}}

= \frac{13}{3 ( 4 x + 1 ) ^{\frac{2}{3}} ( - 5 x + 2 ) ^{\frac{4}{3}}}