derivative of ln((x+sqrt(x^2+4)/2))

解

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

\frac{1}{x ^{2} + 4}

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

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

= \frac{1}{x ^{2} + 4}