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

Lösung

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

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

Schritte zur Lösung

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

Wende die Produktregel an:

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

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

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

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

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

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

Vereinfache

2 x x^{2} + 1 + \frac{x}{x ^{2} + 1} x^{2} : \frac{3 x ^{3} + 2 x}{x ^{2} + 1}

= \frac{3 x ^{3} + 2 x}{x ^{2} + 1}