derivative 8arcsin(e^{x/2})

解

導関数

8 arcsin (e^{\frac{x}{2}})

\frac{4 e ^{\frac{x}{2}}}{1 - e ^{x}}

解答ステップ

\frac{d}{d x} (8 arcsin (e^{\frac{x}{2}}))

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 8 \frac{d}{d x} (arcsin (e^{\frac{x}{2}}))

連鎖法則を適用:

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

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

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

= 8 \cdot \frac{1}{1 - ( e ^{\frac{x}{2}} ) ^{2}} e^{\frac{x}{2}} \frac{1}{2}

簡素化

8 \cdot \frac{1}{1 - ( e ^{\frac{x}{2}} ) ^{2}} e^{\frac{x}{2}} \frac{1}{2} : \frac{4 e ^{\frac{x}{2}}}{1 - e ^{x}}

= \frac{4 e ^{\frac{x}{2}}}{1 - e ^{x}}