derivative y= x/(x+b/x)

Lösung

ableitung von

y = \frac{x}{x + \frac{b}{x}}

\frac{2 b x}{( x ^{2} + b ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{x}{x + \frac{b}{x}})

Wende die Quotientenregel an:

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

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

\frac{d x}{d x} = 1

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

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

Vereinfache

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

= \frac{2 b x}{( x ^{2} + b ) ^{2}}

t a y l or \frac{1}{( 1 + x ) ^{3}}, 0

in v erse l a pl a ce \frac{3}{s ^{2} + 9}

t an g e n t h (x) = \frac{9}{x}, at x = 2

t a y l or e^{- 5 x - 3}, 5

d er i v a t i v e f (x) = (x^{2} - 10 x) \cdot (x - 14)