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

Lösung

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

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

Schritte zur Lösung

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

Wende Exponentenregel an:

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

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

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

Wende die Kettenregel an:

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

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

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

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

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

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

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