derivative of x*sqrt(x^2+1)

Lösung

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

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

Schritte zur Lösung

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

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = x^{2} + 1

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

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

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

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

Vereinfache

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

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