derivative sqrt((4+x^2)/(4-x))

解

導関数

\frac{4 + x ^{2}}{4 - x}

\frac{- x ^{2} + 8 x + 4}{2 x ^{2} + 4 ( - x + 4 ) ^{\frac{3}{2}}}

解答ステップ

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

連鎖法則を適用:

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

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

\frac{d}{d x} (\frac{4 + x ^{2}}{4 - x}) = \frac{- x ^{2} + 8 x + 4}{( 4 - x ) ^{2}}

= \frac{1}{2 \frac{4 + x ^{2}}{4 - x}} \cdot \frac{- x ^{2} + 8 x + 4}{( 4 - x ) ^{2}}

簡素化

\frac{1}{2 \frac{4 + x ^{2}}{4 - x}} \cdot \frac{- x ^{2} + 8 x + 4}{( 4 - x ) ^{2}} : \frac{- x ^{2} + 8 x + 4}{2 x ^{2} + 4 ( - x + 4 ) ^{\frac{3}{2}}}

= \frac{- x ^{2} + 8 x + 4}{2 x ^{2} + 4 ( - x + 4 ) ^{\frac{3}{2}}}