derivative of (ln(x)/(2x))

Lösung

\frac{d}{d x} (\frac{ln ( x )}{2 x})

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

Schritte zur Lösung

\frac{d}{d x} (\frac{ln ( x )}{2 x})

Entferne die Konstante:

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

= \frac{1}{2} \frac{d}{d x} (\frac{ln ( x )}{x})

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

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

\frac{d}{d x} (\frac{6 - 7 x}{2 + x})

x \to 6 - lim (\frac{x ^{6}}{x - 6})

a re a 5 x, 2 x^{2}

\int x^{\frac{7}{2}} ln (x) d x

\int e^{- x} \cdot (- x) d x