(\partial)/(\partial x)((xy)/(x^2+z^2))

Lösung

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

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

Schritte zur Lösung

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

Behandle

y, z

als Konstanten

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

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

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

\frac{\partial}{\partial x} (x) = 1

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

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

Vereinfache

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

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

\frac{d}{d x} (\frac{- 8 x + 2}{3 x + 5})

x \to - 13 lim (- 12)

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

t an g e n t \frac{4}{x}, at x = 3

in v erse l a pl a ce \frac{s ^{2} + 1}{s + 5}