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

解

導関数

ln (x + 1 + x^{2})

\frac{1}{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}{1 + x ^{2}}

= \frac{1}{1 + x ^{2}}