derivative of (x^4+2x(x^3+2x^2+1))

Lösung

\frac{d}{d x} ((x^{4} + 2 x) (x^{3} + 2 x^{2} + 1))

7 x^{6} + 12 x^{5} + 12 x^{3} + 12 x^{2} + 2

Schritte zur Lösung

\frac{d}{d x} ((x^{4} + 2 x) (x^{3} + 2 x^{2} + 1))

Wende die Produktregel an:

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

f = x^{4} + 2 x, g = x^{3} + 2 x^{2} + 1

= \frac{d}{d x} (x^{4} + 2 x) (x^{3} + 2 x^{2} + 1) + \frac{d}{d x} (x^{3} + 2 x^{2} + 1) (x^{4} + 2 x)

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

\frac{d}{d x} (x^{3} + 2 x^{2} + 1) = 3 x^{2} + 4 x

= (4 x^{3} + 2) (x^{3} + 2 x^{2} + 1) + (3 x^{2} + 4 x) (x^{4} + 2 x)

Vereinfache

(4 x^{3} + 2) (x^{3} + 2 x^{2} + 1) + (3 x^{2} + 4 x) (x^{4} + 2 x) : 7 x^{6} + 12 x^{5} + 12 x^{3} + 12 x^{2} + 2

= 7 x^{6} + 12 x^{5} + 12 x^{3} + 12 x^{2} + 2

\frac{d}{d x} (\frac{ln ( x )}{1 + ln ( x )})

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