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

Lösung

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

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

Schritte zur Lösung

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

Wende Exponentenregel an:

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

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

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

\int \frac{7}{( t - 1 ) t ^{2} - 2 t - 48} d t

n = 0 \sum \infty 3^{n - 1}

y^{^{'}} + \frac{x}{x ^{2} - 9} y = 2 x

n = 3 \sum \infty \frac{( ln ( x ) ) ^{n - 1}}{2 ^{n}}

x \to - 9 lim (\frac{x ^{2} - 4}{x - 9})