derivative f(x)=x^x(ln(x)+1)

Lösung

ableitung von

f (x) = x^{x} (ln (x) + 1)

x^{x} (ln (x) + 1)^{2} + x^{x - 1}

Schritte zur Lösung

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

Wende die Produktregel an:

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

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

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

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

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

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

Vereinfache

x^{x} (ln (x) + 1) (ln (x) + 1) + \frac{1}{x} x^{x} : x^{x} (ln (x) + 1)^{2} + x^{x - 1}

= x^{x} (ln (x) + 1)^{2} + x^{x - 1}

d er i v a t i v e f (x) = x e^{4 x}

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