derivative of sqrt((3x+1/(5x^2+1)))

解

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

\frac{- 15 x ^{2} - 10 x + 3}{2 3 x + 1 ( 5 x ^{2} + 1 ) ^{\frac{3}{2}}}

解答ステップ

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

連鎖法則を適用:

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

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

\frac{d}{d x} (\frac{3 x + 1}{5 x ^{2} + 1}) = \frac{- 15 x ^{2} - 10 x + 3}{( 5 x ^{2} + 1 ) ^{2}}

= \frac{1}{2 \frac{3 x + 1}{5 x ^{2} + 1}} \cdot \frac{- 15 x ^{2} - 10 x + 3}{( 5 x ^{2} + 1 ) ^{2}}

簡素化

\frac{1}{2 \frac{3 x + 1}{5 x ^{2} + 1}} \cdot \frac{- 15 x ^{2} - 10 x + 3}{( 5 x ^{2} + 1 ) ^{2}} : \frac{- 15 x ^{2} - 10 x + 3}{2 3 x + 1 ( 5 x ^{2} + 1 ) ^{\frac{3}{2}}}

= \frac{- 15 x ^{2} - 10 x + 3}{2 3 x + 1 ( 5 x ^{2} + 1 ) ^{\frac{3}{2}}}