(\partial)/(\partial x)((x+y)/(1+x^2))

Lösung

\frac{\partial}{\partial x} (\frac{x + y}{1 + x ^{2}})

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

Schritte zur Lösung

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

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

\frac{\partial}{\partial x} (1 + x^{2}) = 2 x

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

Vereinfache

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

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

\frac{d ^{4}}{d x ^{4}} (\frac{1}{1 - x})

\frac{d}{d t} (t + 3)

\frac{\partial}{\partial y} (7 - x^{2} - 2 y^{2})

t an g e n t f (x) = x^{3}, at x = - 3

x \to - 2 lim (x - 2)