derivative 10arcsin(e^{x/2})

Lösung

ableitung von

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

\frac{5 e ^{\frac{x}{2}}}{1 - e ^{x}}

Schritte zur Lösung

\frac{d}{d x} (10 arcsin (e^{\frac{x}{2}}))

Entferne die Konstante:

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

= 10 \frac{d}{d x} (arcsin (e^{\frac{x}{2}}))

Wende die Kettenregel an:

\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}

= 10 \cdot \frac{1}{1 - ( e ^{\frac{x}{2}} ) ^{2}} e^{\frac{x}{2}} \frac{1}{2}

Vereinfache

10 \cdot \frac{1}{1 - ( e ^{\frac{x}{2}} ) ^{2}} e^{\frac{x}{2}} \frac{1}{2} : \frac{5 e ^{\frac{x}{2}}}{1 - e ^{x}}

= \frac{5 e ^{\frac{x}{2}}}{1 - e ^{x}}

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