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

Lösung

ableitung von

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

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

x^{2} - y^{2} + x y \frac{d y}{d x} = 0

\int \frac{3 x}{( 6 - x ^{2} ) ^{3}} d x

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

\frac{d ^{2}}{d t ^{2}} (cos^{2} (t))

\frac{d}{d x} (\frac{ln ( 3 x )}{x})