derivative f(x)=x^3sqrt(x^4+3x^2-2)

Lösung

ableitung von

f (x) = x^{3} x^{4} + 3 x^{2} - 2

\frac{5 x ^{6} + 12 x ^{4} - 6 x ^{2}}{x ^{4} + 3 x ^{2} - 2}

Schritte zur Lösung

\frac{d}{d x} (x^{3} x^{4} + 3 x^{2} - 2)

Wende die Produktregel an:

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

f = x^{3}, g = x^{4} + 3 x^{2} - 2

= \frac{d}{d x} (x^{3}) x^{4} + 3 x^{2} - 2 + \frac{d}{d x} (x^{4} + 3 x^{2} - 2) x^{3}

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

\frac{d}{d x} (x^{4} + 3 x^{2} - 2) = \frac{x ( 2 x ^{2} + 3 )}{x ^{4} + 3 x ^{2} - 2}

= 3 x^{2} x^{4} + 3 x^{2} - 2 + \frac{x ( 2 x ^{2} + 3 )}{x ^{4} + 3 x ^{2} - 2} x^{3}

Vereinfache

3 x^{2} x^{4} + 3 x^{2} - 2 + \frac{x ( 2 x ^{2} + 3 )}{x ^{4} + 3 x ^{2} - 2} x^{3} : \frac{5 x ^{6} + 12 x ^{4} - 6 x ^{2}}{x ^{4} + 3 x ^{2} - 2}

= \frac{5 x ^{6} + 12 x ^{4} - 6 x ^{2}}{x ^{4} + 3 x ^{2} - 2}

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