derivative of (arctan(x)^{1/x})

Lösung

\frac{d}{d x} ((arctan (x))^{\frac{1}{x}})

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

Schritte zur Lösung

\frac{d}{d x} ((arctan (x))^{\frac{1}{x}})

Wende Exponentenregel an:

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

(arctan (x))^{\frac{1}{x}} = e^{\frac{1}{x} l n (a r c t a n (x))}

= \frac{d}{d x} (e^{\frac{1}{x} l n (a r c t a n (x))})

Wende die Kettenregel an:

e^{\frac{1}{x} l n (a r c t a n (x))} \frac{d}{d x} (\frac{1}{x} ln (arctan (x)))

= e^{\frac{1}{x} l n (a r c t a n (x))} \frac{d}{d x} (\frac{1}{x} ln (arctan (x)))

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

= e^{\frac{1}{x} l n (a r c t a n (x))} (- \frac{ln ( arctan ( x ) )}{x ^{2}} + \frac{1}{x arctan ( x ) ( x ^{2} + 1 )})

e^{\frac{1}{x} l n (a r c t a n (x))} = arctan^{\frac{1}{x}} (x)

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

\frac{d}{d x} (\frac{x ^{4} + 4}{x ^{2}})

\int_{0}^{π} sin^{2} (n x) d x

\int 3 (3 x + 9)^{7} d x

n or ma l f (x) = lo g_{2} (2 x^{2} + 14), at x = 5

\frac{d x}{d t} = 2 + 2 x + t + t x