derivative of ln(x^3(x-1)^4)

Lösung

\frac{d}{d x} (ln (x^{3}) (x - 1)^{4})

\frac{3 ( x - 1 ) ^{4}}{x} + 12 ln (x) (x - 1)^{3}

Schritte zur Lösung

\frac{d}{d x} (ln (x^{3}) (x - 1)^{4})

Wende die Produktregel an:

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

f = ln (x^{3}), g = (x - 1)^{4}

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

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

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

= \frac{3}{x} (x - 1)^{4} + 4 (x - 1)^{3} ln (x^{3})

Vereinfache

\frac{3}{x} (x - 1)^{4} + 4 (x - 1)^{3} ln (x^{3}) : \frac{3 ( x - 1 ) ^{4}}{x} + 12 ln (x) (x - 1)^{3}

= \frac{3 ( x - 1 ) ^{4}}{x} + 12 ln (x) (x - 1)^{3}

\int 2 \cdot e^{2 x} d x

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