derivative y=sqrt(x)(x^2+1)^3

Lösung

ableitung von

y = x (x^{2} + 1)^{3}

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

Schritte zur Lösung

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

Wende die Produktregel an:

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

f = x, g = (x^{2} + 1)^{3}

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

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

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

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

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

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

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