derivative of (x/(1+x)^5)

解

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

\frac{5 x ^{4}}{( 1 + x ) ^{6}}

解答ステップ

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

連鎖法則を適用:

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

= 5 (\frac{x}{1 + x})^{4} \frac{d}{d x} (\frac{x}{1 + x})

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

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

簡素化

5 (\frac{x}{1 + x})^{4} \frac{1}{( 1 + x ) ^{2}} : \frac{5 x ^{4}}{( 1 + x ) ^{6}}

= \frac{5 x ^{4}}{( 1 + x ) ^{6}}