derivative y= x/(sqrt(x^2+1))

Lösung

ableitung von

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

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

Schritte zur Lösung

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

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

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

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

Vereinfache

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

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

s im pl i f y 3 x^{2} - \frac{1}{x}

t an g e n t f (x) = x^{2} + 2 x, at x = 34

x \to \infty lim (2^{x})

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

y^{^{'}} - y = x, y (0) = 1