derivative of \sqrt[4]{(3x-2/(x^2+1)})

解

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

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

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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