derivative of (-x/(x^2+y^2))

Lösung

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

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

Schritte zur Lösung

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

Vereinfache

\frac{- x}{x ^{2} + y ^{2}} : - \frac{x}{x ^{2} + y ^{2}}

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

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

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

\int \frac{6 x - 15}{x ^{2} - 3 x} d x

\int (2 sin (x) + x) d x

\int \frac{1}{( 7 - 8 x )} d x

n = 1 \sum \infty 99

k = 1 \sum \infty (- \frac{3}{4})^{k - 1}