derivative of (2x+1^{x^2-1})

Lösung

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

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

Schritte zur Lösung

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

Wende Exponentenregel an:

a^{b} = e^{b l n (a)}

(2 x + 1)^{x^{2} - 1} = e^{(x^{2} - 1) l n (2 x + 1)}

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

Wende die Kettenregel an:

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

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

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

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

e^{(x^{2} - 1) l n (2 x + 1)} = (2 x + 1)^{x^{2} - 1}

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

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