derivative of (2xy/(x+y))

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

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

\frac{d}{d x} (154)

d er i v a t i v e (\frac{4 x ^{2}}{4 - x})^{3}

\frac{d y}{d x} = \frac{x - y}{x + y}

x \to \infty lim (y)

\int \frac{y ^{2}}{x} d x