derivative (sqrt(x))/(3(x^2+1))

Lösung

ableitung von

\frac{x}{3 ( x ^{2} + 1 )}

\frac{- 3 x ^{2} + 1}{6 x ( x ^{2} + 1 ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{x}{3 ( x ^{2} + 1 )})

Entferne die Konstante:

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

= \frac{1}{3} \frac{d}{d x} (\frac{x}{x ^{2} + 1})

Wende die Quotientenregel an:

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

= \frac{1}{3} \cdot \frac{\frac{d}{d x} ( x ) ( x ^{2} + 1 ) - \frac{d}{d x} ( x ^{2} + 1 ) x}{( x ^{2} + 1 ) ^{2}}

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

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

= \frac{1}{3} \cdot \frac{\frac{1}{2 x} ( x ^{2} + 1 ) - 2 x x}{( x ^{2} + 1 ) ^{2}}

Vereinfache

\frac{1}{3} \cdot \frac{\frac{1}{2 x} ( x ^{2} + 1 ) - 2 x x}{( x ^{2} + 1 ) ^{2}} : \frac{- 3 x ^{2} + 1}{6 x ( x ^{2} + 1 ) ^{2}}

= \frac{- 3 x ^{2} + 1}{6 x ( x ^{2} + 1 ) ^{2}}

s l o p e in t erce pt (3, 2), (6, 8)

\int (sin (\frac{x}{2})) d x

a re a sin (x), 3 x, \frac{π}{2}, π

x \to 1 - lim (\frac{x}{x ^{2} - 1})

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