derivative y=(x+1/x)/(2sqrt(x)+1/x)

Lösung

ableitung von

y = \frac{x + \frac{1}{x}}{2 x + \frac{1}{x}}

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

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

d er i v a t i v e f (x) = arctan (\frac{x - 1}{x + 1})

\frac{\partial}{\partial x} (ln (x^{2} y^{2}))

\int \frac{x ^{2} + 1}{x} d x

d er i v a t i v e y = lo g_{x} (e)

n = 0 \sum \infty \frac{1}{n + 12}