derivative of ln(sqrt((x^3/(x^5+1))))

解

\frac{d}{d x} (ln (\frac{x ^{3}}{x ^{5} + 1}))

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

解答ステップ

\frac{d}{d x} (ln (\frac{x ^{3}}{x ^{5} + 1}))

連鎖法則を適用:

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

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

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

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

簡素化

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

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