derivative of x^{1-x}

Lösung

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

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

Schritte zur Lösung

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

Wende Exponentenregel an:

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

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

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

Wende die Kettenregel an:

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

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

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

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

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

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

d er i v a t i v e 11 x

\frac{d}{d x} (4 x - 12)

\frac{d}{d x} (ln (ln (x + 1)))

x \to 1 - lim (\frac{3}{x} - \frac{2}{ln ( x )})

\frac{d}{d t} (A (e^{- t} - e^{- t} t))