derivative of e^{sqrt((1-x/(1+x))})

解

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

- \frac{e ^{\frac{1 - x}{1 + x}}}{- x + 1 ( x + 1 ) ^{\frac{3}{2}}}

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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