derivative of ln(sqrt((x^2/(x+1))))

解

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

\frac{x + 2}{2 x ( x + 1 )}

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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