derivative of (ln(x)^{{p}(x)})

Lösung

\frac{d}{d x} ((ln (x))^{p (x)})

ln^{p (x)} (x) (ln (ln (x)) \frac{d}{d x} (p (x)) + \frac{p ( x )}{x ln ( x )})

Schritte zur Lösung

\frac{d}{d x} ((ln (x))^{p (x)})

Wende Exponentenregel an:

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

(ln (x))^{p (x)} = e^{p (x) l n (l n (x))}

= \frac{d}{d x} (e^{p (x) l n (l n (x))})

Wende die Kettenregel an:

e^{p (x) l n (l n (x))} \frac{d}{d x} (p (x) ln (ln (x)))

= e^{p (x) l n (l n (x))} \frac{d}{d x} (p (x) ln (ln (x)))

\frac{d}{d x} (p (x) ln (ln (x))) = ln (ln (x)) \frac{d}{d x} (p (x)) + \frac{p ( x )}{x ln ( x )}

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

e^{p (x) l n (l n (x))} = ln^{p (x)} (x)

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

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

in v erse l a pl a ce \frac{16}{s ( s + 2 ) ^{4}}

x \to \infty lim (\frac{e ^{- s x}}{s})

\frac{d y}{d x} y = (8 x^{2} - 16 + 16) e^{x}

d er i v a t i v e (1 + 2 x^{2}) (x - x^{2})