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

解

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

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

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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