derivative of (x/(1+sqrt(1-x^2))^n)

Lösung

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

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

Schritte zur Lösung

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

d er i v a t i v e f (x) = \frac{2 x}{1 - x}

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