derivative of (ln(5x)/(5x))

Lösung

\frac{d}{d x} (\frac{ln ( 5 x )}{5 x})

\frac{1 - ln ( 5 x )}{5 x ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{ln ( 5 x )}{5 x})

Entferne die Konstante:

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

= \frac{1}{5} \frac{d}{d x} (\frac{ln ( 5 x )}{x})

Wende die Quotientenregel an:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{1}{5} \cdot \frac{\frac{d}{d x} ( ln ( 5 x ) ) x - \frac{d x}{d x} ln ( 5 x )}{x ^{2}}

\frac{d}{d x} (ln (5 x)) = \frac{1}{x}

\frac{d x}{d x} = 1

= \frac{1}{5} \cdot \frac{\frac{1}{x} x - 1 \cdot ln ( 5 x )}{x ^{2}}

Vereinfache

\frac{1}{5} \cdot \frac{\frac{1}{x} x - 1 \cdot ln ( 5 x )}{x ^{2}} : \frac{1 - ln ( 5 x )}{5 x ^{2}}

= \frac{1 - ln ( 5 x )}{5 x ^{2}}

t a y l or 1. 5^{x}

\int \frac{x}{9 - x ^{2}} d x

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