derivative of (x-ln(1+x)/(x^2))

Lösung

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

\frac{- x ^{2} - 2 x + 2 x ln ( x + 1 ) + 2 ln ( x + 1 )}{x ^{3} ( x + 1 )}

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

\frac{d}{d x} (x^{2}) = 2 x

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

Vereinfache

\frac{( 1 - \frac{1}{1 + x} ) x ^{2} - 2 x ( x - ln ( 1 + x ) )}{( x ^{2} ) ^{2}} : \frac{- x ^{2} - 2 x + 2 x ln ( x + 1 ) + 2 ln ( x + 1 )}{x ^{3} ( x + 1 )}

= \frac{- x ^{2} - 2 x + 2 x ln ( x + 1 ) + 2 ln ( x + 1 )}{x ^{3} ( x + 1 )}

\int x ln (3) d x

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

y^{^{''}} - 12 y^{^{'}} + 36 y = 4 x e^{6 x}

n = 2 \sum \infty e^{3 - 2 n}

\int 3 (3 x + 9)^{3} d x