derivative of x^nln(1/x)

Lösung

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

- n x^{n - 1} ln (x) - x^{n - 1}

Schritte zur Lösung

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

Vereinfache

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

= \frac{d}{d x} (- x^{n} ln (x))

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

= - \frac{d}{d x} (x^{n} ln (x))

Wende die Produktregel an:

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

f = x^{n}, g = ln (x)

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

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

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

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

Vereinfache

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

= - n x^{n - 1} ln (x) - x^{n - 1}

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