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

Lösung

ableitung von

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

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