derivative of (ln(x)/(x+1))

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

Vereinfache

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

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

in v erse l a pl a ce \frac{( 6 s + 3 )}{s ^{2}}

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

\frac{d ^{4}}{d x ^{4}} (\frac{1}{3 x + 1})

\int 2 x^{3} + 4 x + 5 d x

x \to π lim (sin (\frac{11}{6}))