derivative x/(1+x^2)

Lösung

ableitung von

\frac{x}{1 + x ^{2}}

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

Schritte zur Lösung

\frac{d}{d x} (\frac{x}{1 + x ^{2}})

Wende die Quotientenregel an:

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

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

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

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

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

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

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

\int \frac{3 x - 4}{x ^{2} - x - 6} d x

\frac{d}{d x} (6 x - 3 x)

x \to 5 - lim (c x^{2} + 2 x)

a re a y = \frac{x ^{2}}{2} - 4 x + 6, y = \frac{x}{2} - 1

im pl i c i t x^{\frac{1}{2}} + y^{\frac{1}{2}} = 9