(\partial)/(\partial x)((-5x-y)/(y^2+x^2))

Lösung

\frac{\partial}{\partial x} (\frac{- 5 x - y}{y ^{2} + x ^{2}})

\frac{- 5 y ^{2} + 5 x ^{2} + 2 x y}{( y ^{2} + x ^{2} ) ^{2}}

Schritte zur Lösung

\frac{\partial}{\partial x} (\frac{- 5 x - y}{y ^{2} + x ^{2}})

Behandle

y

als Konstante

Wende die Quotientenregel an:

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

= \frac{\frac{\partial}{\partial x} ( - 5 x - y ) ( y ^{2} + x ^{2} ) - \frac{\partial}{\partial x} ( y ^{2} + x ^{2} ) ( - 5 x - y )}{( y ^{2} + x ^{2} ) ^{2}}

\frac{\partial}{\partial x} (- 5 x - y) = - 5

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

= \frac{( - 5 ) ( y ^{2} + x ^{2} ) - 2 x ( - 5 x - y )}{( y ^{2} + x ^{2} ) ^{2}}

Vereinfache

\frac{( - 5 ) ( y ^{2} + x ^{2} ) - 2 x ( - 5 x - y )}{( y ^{2} + x ^{2} ) ^{2}} : \frac{- 5 y ^{2} + 5 x ^{2} + 2 x y}{( y ^{2} + x ^{2} ) ^{2}}

= \frac{- 5 y ^{2} + 5 x ^{2} + 2 x y}{( y ^{2} + x ^{2} ) ^{2}}

in v erse l a pl a ce \frac{8}{s ^{4}}

f^{^{'}} (x) = \frac{x ^{2} + 3}{cos ( x )}

d er i v a t i v e (\frac{10}{3}) t^{2}

d er i v a t i v e f (x) = \frac{1}{6} x^{4} - 6 x^{2} + 1

\frac{d}{d x} (9 x sin (π x))