derivative of sqrt(x/(x^2-8))

解

\frac{d}{d x} (\frac{x}{x ^{2} - 8})

\frac{- x ^{2} - 8}{2 x ( x ^{2} - 8 ) ^{\frac{3}{2}}}

解答ステップ

\frac{d}{d x} (\frac{x}{x ^{2} - 8})

連鎖法則を適用:

\frac{1}{2 \frac{x}{x ^{2} - 8}} \frac{d}{d x} (\frac{x}{x ^{2} - 8})

= \frac{1}{2 \frac{x}{x ^{2} - 8}} \frac{d}{d x} (\frac{x}{x ^{2} - 8})

\frac{d}{d x} (\frac{x}{x ^{2} - 8}) = \frac{- x ^{2} - 8}{( x ^{2} - 8 ) ^{2}}

= \frac{1}{2 \frac{x}{x ^{2} - 8}} \cdot \frac{- x ^{2} - 8}{( x ^{2} - 8 ) ^{2}}

簡素化

\frac{1}{2 \frac{x}{x ^{2} - 8}} \cdot \frac{- x ^{2} - 8}{( x ^{2} - 8 ) ^{2}} : \frac{- x ^{2} - 8}{2 x ( x ^{2} - 8 ) ^{\frac{3}{2}}}

= \frac{- x ^{2} - 8}{2 x ( x ^{2} - 8 ) ^{\frac{3}{2}}}