derivative of x/(x+y/x)

Lösung

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

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

Schritte zur Lösung

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

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

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

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

Vereinfache

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

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

\int cos^{8} (x) sin (x) d x

\int cos (5 t) d t

x \to 1 - lim (4 x)

in v erse l a pl a ce \frac{s}{s + 8}

\frac{\partial}{\partial x} ((x + y) \cdot e^{x + 2 y})